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Modeling the g-factors, hyperfine interaction and optical properties of semiconductor QDs: the atomistic and eight-band $k \cdot p$ approaches

T0 review · 2 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that a corrected eight-band k·p model reproduces atomistic tight-binding results for electron energies and g-factors in InGaAs/GaAs quantum dots, and approximates hyperfine-induced Overhauser fields closely enough to…

desk verdict A careful, honest TB-vs-k.p benchmark with two genuinely useful k.p corrections; the hyperfine validation is internally consistent but needs an external anchor. read the letter →

arxiv 2506.22380 v2 pith:LZ4MNT32 submitted 2025-06-27 cond-mat.mes-hall physics.comp-ph

Krzysztof Gawarecki , Alina Garbiec , Jakub Stanecki , Micha{l} Zieliński This is my paper · ORCID
classification cond-mat.mes-hallphysics.comp-ph
keywords self-assembledquantumdotstight-bindingeight-bandk.pg-factorhyperfineinteractionOverhauserfieldexcitonradiativelifetimeInGaAs/GaAs
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 benchmarks two standard ways of computing the electronic, spin, and optical properties of self-assembled InGaAs/GaAs quantum dots: atomistic $\mathrm{sp}^3d^5s^*$ tight-binding and continuum eight-band $k\cdot p$. Its central claim is that the $k\cdot p$ method, after three targeted corrections, reproduces the atomistic results well enough to serve as a much cheaper design tool: electron and hole single-particle energies agree, electron $g$-factors match closely, and hyperfine-induced Overhauser field fluctuations are approximated with reasonable accuracy. The corrections are a strain-dependent lattice-scaling factor, second-order deformation potentials, and a modified remote-band correction to the $g$-factor that uses a strain-dependent effective gap. The paper also documents where the agreement fails—hole $g$-factors are more sensitive to shear strain, and exciton lifetimes differ by up to 30% at high indium content—so it draws practical criteria for choosing between the two frameworks.

What carries the argument

The argument is carried by the eight-band $k\cdot p$ Hamiltonian in the envelope-function approximation, modified in three places. The lattice-scaling factor $\eta(\mathbf r)$ multiplies the linear-$k$ interband terms and mimics the atomic relaxation that the fixed numerical grid cannot capture; second-order Bir–Pikus deformation potentials improve the strain response of the valence band; and the modified remote-band correction $\bar g'$ replaces the bare reduced Kane energy in the Roth term with a strain-dependent effective gap $\tilde E_g$. For the hyperfine part, the $k\cdot p$ wavefunctions are expanded in hydrogen-like $s$, $p$, and $d$ orbitals with parameters shared with the tight-binding model, and the Overhauser-field fluctuations are computed by projecting the hyperfine Hamiltonian onto the ground Zeeman doublet.

What would settle it

Compute the electron Overhauser-field fluctuations in the same dot geometry using radial matrix elements from an independent first-principles source, or measure the fluctuations directly in a spin-echo experiment; if the independent value shifts by as much as the reported $k\cdot p$--tight-binding difference, the shared hydrogen-like parameterization is responsible for the apparent agreement.

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Extended reading notes

Core claim

The central claim, stated the way a sympathetic reader would state it, is that the eight-band $k\cdot p$ envelope-function model can be upgraded from a qualitative to a near-quantitative tool for spin physics in InGaAs/GaAs quantum dots. The upgrade consists of an effective lattice-scaling factor $\eta(\mathbf r)=1-[a(\mathbf r)-a]/a$ that partially recovers atomistic strain relaxation, second-order Bir–Pikus deformation potentials, and a strain-aware remote-band correction to the electron $g$-factor in which the Roth-type term is evaluated with a strain-modified energy gap. With these changes, the $k\cdot p$ electron and hole energies track the tight-binding results over dot height and indium composition, the electron $g$-factor comes into close agreement, and the hyperfine Overhauser-field fluctuations fall within roughly 200 neV of the atomistic values for electrons at high indium content, with better agreement for holes. The paper takes this as validation that the continuum method, with careful modeling of Bloch functions and orbital character, can approximate atomistic spin-decoherence mechanisms.

Load-bearing premise

The hyperfine comparison checks the two models against each other, but both use the same published hydrogen-like radial exponents and matrix elements for the hyperfine interaction, so a shared error in those numbers would make the agreement look better than it is.

Editorial extensions

If this is right

  • The corrected $k\cdot p$ method can serve as a computationally cheaper first pass for designing dots where electron $g$-factors and single-particle level structure matter, with tight binding reserved for final verification.
  • Because the hyperfine fluctuations agree, $k\cdot p$ can be used to estimate Overhauser-field magnitudes and spin-decoherence times in dots or dot arrays too large for atomistic simulation.
  • The residual hole $g$-factor and exciton-lifetime discrepancies identify concrete regimes—shear-strain-sensitive hole states and high-indium-content emission—where tight binding remains the safer choice.
  • The validated corrections give a concrete parameter recipe for applying the eight-band model to other III-V self-assembled dots with comparable strain profiles.

Reading between the lines

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

  • An implication the paper leaves implicit is that the lattice-scaling factor $\eta$ is a single heuristic parameter; testing it against dots of very different aspect ratios or alloy profiles would show whether it absorbs genuine strain relaxation or merely compensates for other $k\cdot p$ errors.
  • Because both hyperfine implementations share the same hydrogen-like orbital parameterization, the paper establishes internal consistency rather than absolute accuracy; an independent calibration against measured Overhauser fields would be the logical next test.
  • The same benchmarking strategy could be extended to other spin-optical observables, such as spin-flip rates or dynamic nuclear polarization profiles, where the continuum model's approximate Bloch functions may or may not remain adequate.
  • A practical extension would be to turn the corrected $k\cdot p$ model into a fast screening tool for dot arrays: the roughly 200 neV electron Overhauser discrepancy at high indium content suggests the continuum model slightly underestimates electron wavefunction spread, a bias that could be corrected systematically.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper presents a systematic comparison of atomistic sp3d5s* tight-binding (TB) and continuum eight-band k·p methods for modeling self-assembled InGaAs/GaAs quantum dots. It covers single-particle energy levels, electron and hole g-factors, exciton radiative lifetimes, and hyperfine-induced Overhauser field fluctuations. To improve the k·p description, the authors introduce three targeted corrections: second-order deformation potentials, a modified remote-band correction for g-factors, and an effective lattice-scaling scheme. They report improved agreement with TB for electron energies and g-factors, persistent discrepancies for hole g-factors, and lifetime differences of up to 30%. The hyperfine interaction is implemented within both frameworks and the resulting Overhauser field fluctuations are compared, with the paper claiming reasonable agreement that validates the k·p approach as a computationally cheaper tool for spin-related QD studies.

Significance. If the claims hold, the paper provides a valuable systematic benchmark between two widely used modeling frameworks and offers practical improvements to the eight-band k·p method. The work is strengthened by thorough numerical comparisons across multiple observables, internal cross-checks such as the agreement between position- and momentum-based oscillator strengths in TB, and a documented procedure for extracting bulk parameters from TB in Appendix A3. The proposed k·p corrections contain no numerically fitted constants in the strict sense. However, the hyperfine validation is weakened by the fact that both models share the same hydrogen-like radial parameters taken from Ref. [33], so the absolute hyperfine scale is not independently tested. This limits the strength of the conclusion that the k·p method 'can approximate atomistic spin decoherence mechanisms with reasonable accuracy' as an atomistic statement, although the comparison still tests the distribution of orbital weights over cation and anion sites.

major comments (2)
  1. [Sec. VI, Appendix A5, Fig. 5] The claim that the TB model provides an atomistic benchmark for the hyperfine interaction is weakened by the fact that both the TB and k·p implementations use the same hydrogen-like radial functions with parameters ξ_S, ξ_P, ξ_D and M_ll' taken from Ref. [33]. As stated in Appendix A5, the TB basis is represented by these hydrogen-like orbitals, so the absolute scale of the Overhauser fluctuations (∝ |R_S(0)|^2 ∝ ξ_S^3 and M_ll') is set by the same shared inputs. The agreement in Fig. 5 therefore tests how similarly the two methods distribute orbital weights over cation and anion sites, but it does not validate the absolute hyperfine couplings or the claim in Sec. VIII that the k·p method 'can approximate atomistic spin decoherence mechanisms with reasonable accuracy.' To support that claim, the authors should either calibrate ξ and M_ll' against experimental hyperfine constants or ab initio calculations, or explicitly reframe the validation as an internal consistency check between the two model implementations.
  2. [Sec. IV.D, Eq. (11)] The modified remote-band correction for the electron g-factor introduces the strain-corrected gap ẽ_g, with the paper noting that an alternative choice (the QD electron-hole energy difference) would also be reasonable. Because the improvement of electron g-factors is a central result of the paper, the sensitivity of the corrected g-factors to this choice should be assessed; without such an analysis, it is unclear whether the improvement is robust or specific to the particular definition of ẽ_g.
minor comments (5)
  1. [Table I] The lattice constants in Table I appear to be interchanged between the InAs and GaAs columns: the standard values are a(InAs) = 6.0583 Å and a(GaAs) = 5.6535 Å, while the table lists the opposite.
  2. [Appendix A3] The units of E_P^(red) (presumably eV) are missing; the values 20.5 and 19.5 should be labeled explicitly.
  3. [Section III] In the sentence 'The rational behind this approach', 'rational' should be 'rationale'.
  4. [Appendix A5] Given the central role of the hyperfine parameters ξ_S, ξ_P, ξ_D and M_ll', the authors should reproduce these values in an appendix or supplementary material rather than only referring to Ref. [33].
  5. [Section VII.C] The statement that the tight-binding position- and momentum-based oscillator strengths agree 'with a few percent of difference' would benefit from a quantitative statement of the range of deviations observed in Fig. 4.

Circularity Check

2 steps flagged · score 4.0 of 10

The TB hyperfine benchmark inherits the same hydrogenic radial parameters from the authors' earlier k.p paper, and the k.p Hamiltonian is parametrized from the same TB model used for validation, so the comparison is internally consistent but not an independent atomistic test.

  1. ansatz smuggled in via citation [Sec. VI and Appendix A5 (hyperfine interaction); Fig. 5]
    "In contrast to the k·p formulation, the tight-binding approach provides information about the wave function localization on individual atomic nodes. ... Finally, the tight-binding basis was represented by hydrogen-like orbitals [R_α(r) → R_nl(r)] ... The values of ξ_S, ξ_P, ξ_D exponents and M_ll' for the relevant atoms, are given in Ref. [33]."

    The paper's central claim is that the eight-band k.p method 'can approximate atomistic spin decoherence mechanisms,' and this is validated by benchmarking against the sp3d5s* tight-binding model in Fig. 5. However, the TB hyperfine calculation does not use radial orbitals inherent to the empirical TB basis; it replaces them with the same hydrogen-like R_nl(r) functions and the same ξ_S, ξ_P, ξ_D and M_ll' values taken from Ref. [33], the earlier k.p hyperfine paper co-authored by the first author. The contact term scales as |R_S(0)|^2 ∝ ξ_S^3 and the orbital/dipolar terms are set by M_ll', so any error in these shared inputs shifts both the k.p and TB Overhauser curves in the same direction and cancels in the comparison.

  2. fitted input called prediction [Appendix A3; Sec. VII.A-B; Sec. VIII]
    "The material parameters m*e and γ1−3 for the 8-band k·p model are extracted from the bulk band structures of the tight-binding model, where we took the second derivatives (in the [001] and [111] directions) of the band energies at k = 0. ... The bulk g-factors for: the electron (g), the heavy-hole (g_hh), and the light-hole (g_lh) are calculated from the TB within the linear response theory [8, 19]."

    The eight-band k.p model is validated against the same tight-binding model that supplies its bulk input parameters: the effective mass and Luttinger parameters are extracted from the TB bulk dispersion, and the bulk g-factor-derived κ and q are computed from TB linear-response calculations. Consequently, the close agreement in the single-particle energies (Fig. 2) and the improvement in electron g-factors (Fig. 3) are partly forced in the bulk limit by construction: the k.p Hamiltonian was tuned to reproduce the TB band structure and Zeeman response. The QD-level comparison retains nontrivial content because strain, confinement, and the proposed corrections are not fitted to the TB dot data, so the circularity is partial rather than total.

full rationale

Two genuine circularity burdens are identifiable in the text, and both are acknowledged by the authors. First, the eight-band k.p model is not parameter-free relative to the benchmark: its effective masses, Luttinger parameters, and g-factor-derived κ and q are extracted from the same tight-binding model against which the k.p results are subsequently validated. The agreement in Sec. VII.A-B therefore contains a fitted, by-construction component in the bulk regime. Second, and more consequential for the paper's strongest claim, the TB hyperfine benchmark is not an independent atomistic reference: Appendix A5 replaces the TB orbitals by hydrogen-like functions and takes the radial exponents and matrix elements from Ref. [33], the earlier k.p hyperfine paper co-authored by the first author. Both the k.p and TB Overhauser-field calculations thus share the same dominant radial scale and matrix elements, so the Fig. 5 agreement can validate how similarly the two methods distribute orbital weight over cation and anion sites, but it cannot establish the absolute atomistic accuracy of the hyperfine coupling. The non-hyperfine portions of the paper, including the strain model, the η lattice-scaling scheme, the modified remote-band correction of Eq. (11), and the exciton-lifetime comparisons, have independent content and are not numerically fitted to the target TB results. For this reason the paper is not wholly circular, but the principal 'atomistic validation' claim is weakened by a self-cited shared input, giving a moderate score of 4 rather than a higher one.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claims rest on the standard machinery of both models plus several parameter choices. The k.p model inherits reduced Kane energies (20.5/19.5 eV) to avoid spurious solutions, and its bulk parameters are extracted from the TB model's bulk band structure, so the two methods are not fully independent at the input level. The hyperfine comparison in both models uses hydrogen-like orbital parameters from the authors' earlier Ref. [33]. The VFF strain model uses reduced elastic constants for InAs. No physically new entities are introduced.

free parameters (4)
  • Reduced Kane energy E_P^(red) = 20.5 eV (GaAs), 19.5 eV (InAs)
    The physical E_P values produce spurious k.p solutions, so reduced values are used (Appendix A3); the reduction shifts energies and g-factors and is compensated by the modified remote-band term in Eq. 11.
  • Strain-corrected gap ẽ_g in the remote-band correction = E_g + (a_c - a_v)(ε_xx + ε_yy + ε_zz)
    Eq. 11's correction term uses a chosen strain-modified gap; the paper notes a self-consistent QD gap is an alternative, so this is a modeling choice that sets the electron g-factor correction scale.
  • Lattice scaling factor η(r) = η(r) = 1 - (a(r) - a)/a, no fitted constant
    Ad hoc form introduced in Sec. IVC to absorb strain-induced atomic displacements in the linear-k interband terms; its validation is the improved agreement with TB it produces.
  • Hydrogen-like hyperfine radial parameters (ξ_S, ξ_P, ξ_D, M_ll') = values taken from Ref. [33]
    Empirical orbital exponents used in both the k.p and TB hyperfine matrix elements (Appendix A5); both models share these values, coupling their Overhauser field results.
assumptions (5)
  • domain assumption Eight-band k.p truncation with perturbative remote-band terms suffices for InGaAs/GaAs dots
    Entire k.p framework of Sec. IVA assumes Γ6c-Γ8v-Γ7v bands plus remote-band corrections capture the physics.
  • domain assumption Covalent VFF strain model with reduced C12 and C44 for InAs gives reliable atomic positions
    Appendix A1 uses the covalent approximation and reduced elastic constants because the full Martin model is not used; strain drives energies, g-factors, and lifetimes.
  • domain assumption TB orbitals can be treated as spherical harmonics for the orbital Zeeman term
    Sec. III and Appendix A2 state this approximation explicitly; it enters the orbital angular momentum matrices used in g-factor calculations.
  • domain assumption Hydrogen-like radial functions from Ref. [33] represent the radial parts of TB orbitals in hyperfine matrix elements
    Appendix A5 replaces the TB radial parts with hydrogen-like functions; the exponents come from a self-cited paper and are shared with the k.p model.
  • domain assumption Inter-atomic contributions to position matrix elements can be neglected (Eq. 14)
    The position-operator route to oscillator strengths drops local inter-atomic terms; the paper checks this by comparing with the momentum route, which agrees within a few percent in TB.

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Cite this review

Pith. "Pith review of Modeling the g-factors, hyperfine interaction and optical properties of semiconductor QDs: the atomistic and eight-band $k \cdot p$ approaches." pith.science (2026). https://pith.science/paper/LZ4MNT32

@misc{pith2026250622380,
  author       = {Pith},
  title        = {Pith review of: Modeling the g-factors, hyperfine interaction and optical properties of semiconductor QDs: the atomistic and eight-band $k \cdot p$ approaches},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LZ4MNT32}},
  note         = {Machine review of arXiv:2506.22380}
}
abstract

We present a detailed comparative study of two important theoretical approaches: atomistic sp$^3$d$^5$s$^*$ tight-binding and continuum eight-band $k \cdot p$ methods, for modeling the spin and optical properties of quantum dots (QDs). Our investigation spans key physical observables, including single-particle energy levels, g-factors, exciton radiative lifetimes, and hyperfine-induced Overhauser field fluctuations. We perform our calculations for self-assembled InGaAs/GaAs QD systems as representative case studies. While both methods yield qualitatively consistent trends, quantitative discrepancies arise due to different treatment of atomistic details, strain effects, and confinement. We introduce targeted corrections to the eight-band $k \cdot p$ framework, including a modified deformation potential scheme and adjusted remote-band contributions, to improve agreement with atomistic results, especially for electron g-factors and single-particle energies. Furthermore, we validate the eight-band implementation of hyperfine interactions by benchmarking it against the tight-binding model, showing reasonable convergence for both electrons and holes. Our results establish criteria for selecting the appropriate modeling framework based on the desired physical accuracy and computational efficiency in spin-optical studies of semiconductor QDs.

Figures

Figures reproduced from arXiv: 2506.22380 by the authors.

Figure 1
Figure 1. FIG. 1. Composition distribution for the InGaAs/GaAs QD. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The electron and hole energy dependence on the QD height (a,b) and the QD composition (c,d). [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The electron and hole g-factor dependence on the QD height (a,b) and the QD composition (c,d). [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The radiative lifetime dependence on the QD height (a,b) and the QD composition (c,d). For the blue points, the oscillator strength is [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The fluctuations of the longitudinal and the transverse Overhauser field components calculated for the ground Zeeman doublet, for [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

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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

  1. Light-hole states and hyperfine interaction in electrically-defined Ge/GeSn quantum dots

    cond-mat.mes-hall 2025-07 conditional novelty 7.0 of 10

    In tensile-strained Ge/GeSn light-hole qubits, hyperfine coupling is dominated by the Fermi contact term via s-type conduction-band admixtures, and it grows with Sn barrier concentration.

Reference graph

Works this paper leans on

89 extracted references · 62 canonical work pages · cited by 1 Pith paper

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    S.Schulz,S.Schumacher,andG.Czycholl,Tight-bindingmodel for semiconductor quantum dots with a wurtzite crystal struc- ture: Fromone-particlepropertiestoCoulombcorrelationsand optical spectra, Phys. Rev. B73, 245327 (2006)

    work page 2006

  2. [1]

    Here the indices𝑛, 𝑚 go over atoms; and𝛼, 𝛽 over the orbitals

    The oscillator strength of the exciton state|𝑋𝑛⟩ is cal- culated from 𝑓𝑛= 2𝑚0 ℏ2𝑒2𝐸(X) 𝑛 ∑︁ 𝜇=𝑥,𝑦,𝑧 ⟨vac.|𝐷𝜇|𝑋𝑛⟩ 2, where 𝑫= ∑︁ 𝑖,𝑗 𝒅𝑖𝑗 ℎ𝑖𝑎𝑗, giving 𝑓𝑛= 2𝑚0 ℏ2𝑒2𝐸(X) 𝑛 ∑︁ 𝜇=𝑥,𝑦,𝑧 ∑︁ 𝑖,𝑗 𝑐(𝑛) 𝑗𝑖 𝑑𝜇 𝑖𝑗 2 , (13) where 𝒅𝑖𝑗 = −𝑒𝒙𝑖𝑗 with the matrix elements of the position operator 𝒙𝑖𝑗 = ∑︁ 𝑛,𝑚 ∑︁ 𝛼,𝛽 𝑤(𝑖)∗ 𝑛,𝛼𝑤(𝑗) 𝑚,𝛽⟨𝑹𝑛;𝛼|𝒙| 𝑹𝑚;𝛽⟩. Here the indices𝑛, 𝑚 go ove...

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    (16) Inthecalculationof 𝒙𝑖𝑗 and 𝒑𝑖𝑗,thevalence-bandstatesare represented in the electron picture – not in the hole picture as in the other parts of the paper

    Theoscillatorstrengthof |𝑋𝑛⟩iscalculatedfrom[28,61] 𝑓𝑛= 2 𝑚0𝐸(X) 𝑛 ∑︁ 𝜇=𝑥,𝑦,𝑧 ∑︁ 𝑖,𝑗 𝑐(𝑛) 𝑗𝑖 𝑝𝜇 𝑖𝑗 2 , (15) The 𝒑𝑖𝑗 momentum matrix elements are calculated us- ingtheHellman-Feynmanntheorem[19,62,63],which gives 𝒑𝑖𝑗 = 𝑖𝑚0 ℏ ∑︁ 𝑛,𝑚 ∑︁ 𝛼,𝛽 𝑤(𝑖)∗ 𝑛,𝛼𝑤(𝑗) 𝑚,𝛽( 𝑹𝑚− 𝑹𝑛)𝑡(𝑛𝑚) 𝛼𝛽 . (16) Inthecalculationof 𝒙𝑖𝑗 and 𝒑𝑖𝑗,thevalence-bandstatesare represented in the el...

  4. [3]

    The eight-band 𝑘 ·𝑝 The material parameters𝑚∗ e and 𝛾1−3 for the 8-band𝑘 ·𝑝 modelareextractedfromthebulkbandstructuresofthetight- 12 binding model, where we took the second derivatives (in the

  5. [4]

    𝜃”, “rr”, “r𝜃

    Strain IntheMartin’sformulationoftheValenceForceField(VFF) model, the elastic energy takes the form [34, 35] 𝑈M= ∑︁ 𝑖 NN(𝑖)∑︁ 𝑗 ( 1 4𝑘(r) 𝑖𝑗 𝑟𝑖𝑗−𝑑𝑖𝑗 2 + ∑︁ 𝑘≠𝑖,𝑘>𝑗 1 2𝐾(𝜃) 𝑖𝑗𝑘 𝑑𝑖𝑗𝑑𝑖𝑘 𝜃𝑖𝑗𝑘 −𝜃(0) 𝑖𝑗𝑘 2 +𝐾(rr) 𝑖𝑗𝑘 𝑟𝑖𝑗−𝑑𝑖𝑗 (𝑟𝑖𝑘−𝑑𝑖𝑘) +𝐾(r𝜃) 𝑖𝑗𝑘 𝑑𝑖𝑗 𝑟𝑖𝑗−𝑑𝑖𝑗 +𝑑𝑖𝑘(𝑟𝑖𝑘−𝑑𝑖𝑘) 𝜃𝑖𝑗𝑘 −𝜃(0) 𝑖𝑗𝑘 ) , where NN(𝑖) are the nearest neighbors of the𝑖-th atom,𝑟𝑖𝑗 = 𝒓𝒊 𝒋 with 𝒓𝑖...

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    The tight-binding model The tight-binding Hamiltonian𝐻TB is originally written in the real basis [38] of 𝑠, 𝑝𝑥, 𝑝𝑦, 𝑝𝑧, 𝑑𝑥𝑦, 𝑑𝑦𝑧, 𝑑𝑧𝑥, 𝑑𝑥2−𝑦2, 𝑑3𝑧2−𝑟2, 𝑠∗ orbitals with two spin configurations. In this basis, the orbital angular momentum on-site matrices (for a given spin) take the form 𝐿𝑥 =𝑖ℏ ©­­­­­­­­­­­­­­ « 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ...

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    Vrijen and E

    R. Vrijen and E. Yablonovitch, A spin-coherent semiconduc- tor photo-detector for quantum communication, Phys. E: Low- Dimens. Syst. Nanostructures10, 569 (2001)

    2001

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    To avoid spurious solutions [50] in diagonalizing the𝑘 ·𝑝 Hamiltonian, the𝐸P is reduced to:𝐸(red) P = 20.5 for GaAs, and𝐸(red) P = 19.5 for InAs

    and [111] directions) of the band energies at𝒌 = 0. To avoid spurious solutions [50] in diagonalizing the𝑘 ·𝑝 Hamiltonian, the𝐸P is reduced to:𝐸(red) P = 20.5 for GaAs, and𝐸(red) P = 19.5 for InAs. The values of the𝐴′ and𝛾′ 1−3 are calculated from [47] 𝐴′=𝑚0 𝑚∗e − " 2 3 𝐸(red) P 𝐸g + 1 3 𝐸(red) P 𝐸g+Δ0 # , 𝛾′ 1=𝛾1− 1 3 𝐸(red) P 𝐸g , 𝛾′ 2=𝛾2− 1 6 𝐸(red) P ...

Show all 89 references
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    The calculations can be optimized by moving to the reciprocal space

    The Coulomb matrix elements and lifetime calculations The electron-hole direct Coulomb matrix elements are de- fined by 𝑉𝑖𝑗𝑗 ′𝑖′ = |𝑒|2 4𝜋𝜖0𝜖𝑟 ∫ d𝒓 ∫ d𝒓′ × 𝜓(e)∗ 𝑖 (𝒓)𝜓(h)∗ 𝑗 (𝒓′)𝜓(h) 𝑗′ (𝒓′)𝜓(e) 𝑖′ (𝒓) |𝒓− 𝒓′| , where𝜓(e) 𝑖 (𝒓) and𝜓(h) 𝑖 (𝒓) are electron and hole wave func- t...

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