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REVIEW 2 major objections 6 minor 20 references

Modelling the mean inner potential of alloyed and strained materials

T0 review · 2 major / 6 minor · reviewed 2026-07-08 · glm-5.2

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Pith's one-line read Two-rule model predicts mean inner potential of alloys and strained crystals

desk verdict Practical MIP model for alloys/strain: useful, validated, but theoretical justification is muddled read the letter →

arxiv 2607.05948 v1 pith:ZABGUAZC submitted 2026-07-07 cond-mat.mtrl-sci

Marco Schowalter , Peer Kruse , Andreas Rosenauer This is my paper · ORCID
classification cond-mat.mtrl-sci
keywords meaninnerpotentialalloymodelingstrainvolumerescalinglinearinterpolationdensityfunctionaltheoryelectronholographymuffin-tinapproximation
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 mean inner potential (MIP) — the average electrostatic potential inside a crystal, relevant to electron holography — can be predicted for alloyed and strained materials using only the MIPs and lattice parameters of the pure endpoint materials. The authors argue that two effects dominate: (1) at constant volume, the MIP of an alloy A_xB_{1-x} is a simple linear combination of the endpoint MIPs, and (2) when volume changes (due to lattice mismatch, strain, or density variation), the MIP rescales proportionally as the ratio of old to new volume. Combining these two rules reproduces DFT-computed MIPs for AlGaAs, InGaP, and GeSi alloys, for biaxially strained GeSi, for uniaxially/biaxially/hydrostatically strained GaP, and even for the density dependence of amorphous carbon, with deviations below ~2.2% — below the detection limit of current electron holography. The physical reason the model works is that the largest contribution to the MIP comes from the Coulomb potential deep within atomic cores (muffin-tin spheres), which is insensitive to bonding environment, while the integral of the potential stays approximately constant when the cell deforms, so the MIP changes mainly because the dividing volume changes.

What carries the argument

Mean inner potential (MIP): volume-averaged Coulomb potential of a crystal. Eq. 3: V_0(A_xB_{1-x}) = x*V_0(A) + (1-x)*V_0(B), linear interpolation at constant volume. Eq. 4: V'_0(Ω') = (Ω/Ω')*V_0(Ω), volume rescaling. Modified atomic scattering amplitudes (MASAs): DFT-derived scattering factors that account for charge redistribution, whose forward-scattering extension underpins Eq. 3. Muffin-tin spheres: atomic-core regions whose Coulomb potential dominates the MIP and is insensitive to bonding.

What would settle it

A material system where bonding-induced charge redistribution contributes comparably to or more than the atomic-core potential — e.g., a highly ionic alloy with large charge transfer — would show MIPs that deviate from the two-rule prediction by more than the ~2.2% maximum deviation seen here, falsifying the model's claim of universal applicability.

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Extended reading notes

Core claim

The paper establishes that the MIP of a mixed or strained crystal factorizes into a composition-dependent linear interpolation (Eq. 3) and a volume-dependent rescaling (Eq. 4), and that this two-equation model captures the bowing in MIP-vs-concentration curves that pure linear interpolation misses. The central object is the MIP itself, defined as the volume average of the crystal Coulomb potential (Eq. 1), and the central mechanism is that this average is dominated by atomic-core contributions that are nearly invariant under alloying, leaving volume change as the primary correction. The authors validate this against full DFT calculations for three alloy systems (one with negligible lattice-m

Load-bearing premise

The model assumes that charge redistribution due to alloying, straining, or surface relaxation is small enough to neglect, so that the MIP is dominated by atomic-core potentials that are insensitive to bonding environment. If charge redistribution is not negligible, both the linear interpolation and the volume rescaling lose accuracy.

Editorial extensions

If this is right

  • Researchers using electron holography on alloyed or strained semiconductor devices can estimate MIPs without running full DFT supercell calculations, using only tabulated endpoint values and lattice parameters.
  • The model provides a consistency check: if measured MIPs deviate significantly from the two-rule prediction, the discrepancy may signal charge redistribution effects large enough to matter — a diagnostic rather than a nuisance.
  • The volume-rescaling rule (Eq. 4) is material-agnostic and could be tested on any system where density or pressure changes the MIP, including high-pressure phases or porous materials.
  • The finding that covalent and ionic alloys behave similarly under the model suggests that bonding character is secondary to atomic-core dominance for MIP, which constrains future charge-redistribution corrections.

Reading between the lines

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

  • If the model holds for ternary or quaternary alloys (e.g., InGaAsP), the MIP would be a weighted sum of endpoint MIPs with a single volume rescaling — a direct extension the paper does not test.
  • The model's assumption that charge redistribution is negligible relative to the isolated-atom baseline could break down for materials with strong charge transfer (e.g., highly ionic compounds or metals with delocalized electrons), where the muffin-tin-sphere dominance may not hold.
  • The ~0.76 V y-intercept found in prior amorphous-carbon data, which the model forces to zero, may indicate a systematic offset from surface dipoles or reference-potential conventions rather than a true density-independent contribution — worth separating experimentally.
  • For nanoscale objects where surface-to-volume ratios are large, the surface-potential contribution (noted as ~0.2 V by Kim et al.) could become comparable to volume-rescaling effects, setting a size scale below which the model needs correction.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 6 minor

Summary. This manuscript proposes a simple two-equation model for the mean inner potential (MIP) of alloyed and strained materials: Eq. (3) gives a linear combination of endpoint MIPs for alloying at constant volume, and Eq. (4) rescales the MIP by the volume ratio when the cell volume changes. The model requires no fitted parameters—only the MIPs and lattice parameters of the pure endpoint materials. It is validated against full DFT (WIEN2k, APW+LO) calculations for strained GaP (uniaxial, biaxial, hydrostatic), and for AlGaAs, InGaP, and GeSi alloys, with reported deviations of 1–2%, below typical electron-holography detection limits. The model is also tested against literature data for amorphous carbon density dependence and for biaxially strained GeSi.

Significance. The paper addresses a practical problem in electron microscopy: estimating MIPs of alloys and strained materials without performing full DFT calculations for every composition and strain state. The model is parameter-free and falsifiable, and the authors provide DFT validation across multiple material systems (III-V and group-IV semiconductors) and strain conditions. Data is deposited on Zenodo. The 1–2% agreement with full DFT is a useful result for the electron-holography community, where MIP values are needed for phase interpretation. The contribution is primarily empirical and practical rather than theoretically novel, but it is well-scoped for the intended audience.

major comments (2)
  1. The theoretical derivation of Eqs. (3) and (4) is stated to rely on the MASA (modified atomic scattering amplitudes) concept for forward scattering, but MASAs for forward scattering are never actually computed. The phrase 'Assuming that we would have computed MASAs for forward scattering' appears twice (items 1 and 2 in the key-idea list). This is a genuine gap in the theoretical justification. However, the authors' own explanation in the final paragraph—that linearity arises because 'the largest contribution to the MIP arises from the large Coulomb potential within the muffin-tin spheres and thus, becomes linear by definition'—is a more direct and self-contained justification that does not require MASAs at all. The authors should either (a) compute the forward-scattering MASAs to close the loop on the stated derivation, or (b) reframe the theoretical justification around the muffin-tin/
  2. The muffin-tin dominance argument, which the authors identify as the actual reason the model works (final paragraph: 'about 6% for GaP'), is quantified only for GaP. The DFT validation covers AlGaAs, InGaP, and GeSi, but the interstitial fraction of the MIP is not reported for any of these systems. For GeSi in particular, which is more covalent, the interstitial contribution could differ. Since the muffin-tin transferability assumption is load-bearing for explaining why the model succeeds across all tested systems, the authors should provide the muffin-tin/interstitial decomposition for at least one additional system (ideally GeSi) to confirm that the mechanism is consistent. If the decomposition is not readily available, this limitation should be explicitly acknowledged.
minor comments (6)
  1. Eq. (2): the notation uses a superscript 'iso' on V_0 that is not defined in the surrounding text; please clarify that this refers to the isolated-atom approximation.
  2. Fig. 1 caption: 'Linear relation are found' should read 'Linear relations are found'. Also, 'can be models reasonably' should be 'can be modeled reasonably'.
  3. The text states maximum deviations of '1.1% (0.17 V), 0.8% (0.14 V) and 2.2% (0.28 V)' for the three strain types in GaP. It would help to state whether these are deviations of the model (Eq. 4) from the linear fit, or from the raw DFT data.
  4. Table I lists elastic moduli only for Si and Ge, with dashes for AlAs, GaP, and InP. If these were not needed for the calculations presented, a footnote explaining this would avoid confusion.
  5. The amorphous carbon comparison (final section before summary) compares the model's slope of 5.43 V/(g/cm³) to a refit value of 5.47 V/(g/cm³). The original fit value (5.20 V/(g/cm³)) is also given but the comparison would be clearer if all three values were tabulated.
  6. Reference [13] and [14] appear to be conference proceedings (BIO Web Conf.); the authors may wish to confirm these are the most appropriate citations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for a careful reading and constructive comments. Both major points are well-taken. We agree that the MASA framing is incomplete as presented and that the muffin-tin argument is the more self-contained justification. We will reframe accordingly. We also agree that the muffin-tin/interstitial decomposition should be provided for at least one additional system beyond GaP, and we will compute and report it for GeSi.

read point-by-point responses

  1. Referee: The theoretical derivation of Eqs. (3) and (4) is stated to rely on the MASA concept for forward scattering, but MASAs for forward scattering are never actually computed. The phrase 'Assuming that we would have computed MASAs for forward scattering' appears twice. The authors should either (a) compute the forward-scattering MASAs to close the loop, or (b) reframe the theoretical justification around the muffin-tin argument.

    Authors: The referee is correct. The MASA concept is invoked as motivation but is not carried through to an actual computation for forward scattering, and the conditional phrasing ('Assuming that we would have computed MASAs...') makes this gap explicit rather than hiding it. We agree that option (b) is the more appropriate fix: the muffin-tin dominance argument given in the final paragraph is self-contained and does not require the MASA framework at all. The key observation is that the MIP, defined as the volume average of the Coulomb potential in Eq. (1), receives its dominant contribution from the large Coulomb potential inside the muffin-tin spheres around each atomic site. Within these spheres, the potential is approximately atomic-like, so the integral over each sphere is approximately transferable between the pure material and the alloy. The interstitial contribution, which is where charge redistribution due to bonding would manifest, is small (about 6% for GaP). This directly explains why Eq. (3) (linear interpolation at constant volume) and Eq. (4) (volume rescaling) work: the total integral of the Coulomb potential is approximately conserved under alloying at fixed volume, and the MIP changes mainly through the 1/Ω prefactor when the volume changes. We will rewrite the theoretical justification section to lead with this argument and remove the conditional MASA framing from the key-idea list. We will retain a brief reference to the MASA concept as historical context for the idea that muffin-tin and interstitial contributions can be separated, but we will no longer present it as a load-bearing step in the derivation. revision: yes

  2. Referee: The muffin-tin dominance argument is quantified only for GaP (~6% interstitial). The DFT validation covers AlGaAs, InGaP, and GeSi, but the interstitial fraction is not reported for these. For GeSi in particular, which is more covalent, the interstitial contribution could differ. The authors should provide the muffin-tin/interstitial decomposition for at least one additional system (ideally GeSi), or explicitly acknowledge the limitation.

    Authors: This is a fair point. The transferability of the muffin-tin dominance argument across all tested systems is an assumption that we currently support with data from GaP only. GeSi is the most important test case because it is the most covalent system in our validation set, and one would expect a larger interstitial contribution there. We will compute the muffin-tin/interstitial decomposition of the MIP for both pure Ge and pure Si (and, if feasible, for an intermediate GeSi composition) using the same WIEN2k APW+LO framework. We expect the interstitial fraction to be somewhat larger than 6% for GeSi but still a minority contribution, which would be consistent with the model working well for this system. If the decomposition turns out to show a substantially larger interstitial fraction than expected, we will report this honestly and discuss its implications for the model's transferability. In any case, we will add a sentence to the revised manuscript explicitly stating the limitation that the decomposition has been checked for a limited set of systems. revision: yes

Circularity Check

2 steps flagged · score 3.0 of 10

Model equations are near-definitional consequences of Eqs. 1–2 under a stated assumption; DFT validation provides independent content; self-citations present but not load-bearing for the actual model.

  1. self definitional [Summary paragraph (final paragraph), discussing Eq. 3]
    "This is due to the fact, that the largest contribution to the MIP arises from the large Coulomb potential within the muffin-tin spheres and thus, becomes linear by definition."

    The paper's own summary states that Eq. 3 (linear interpolation of MIPs with concentration) is linear 'by definition.' Indeed, Eq. 2 defines V_0 as (1/Ω) Σ f_i(0); for an alloy A_xB_{1-x}, this sum is x·f_A(0) + (1-x)·f_B(0) by construction of the weighted average, assuming transferable scattering amplitudes. The 'model' Eq. 3 is thus a direct algebraic consequence of Eq. 2 under the stated assumption of negligible charge redistribution, not an independent derivation from the MASA concept.

  2. self definitional [Summary paragraph (final paragraph), discussing Eq. 4]
    "the integral over the Coulomb potential (eq. (1)) stays approximately constant and thus, the MIP is mainly changed by the change in volume."

    Eq. 4 (V'_0 = (Ω/Ω')·V_0) is presented as a model, but the paper's own summary explains it as a direct consequence of Eq. 1: if the integral ∫V d³r stays approximately constant, then V_0 = (1/Ω)·∫V d³r scales as 1/Ω by definition. The 'prediction' is the definition under the stated assumption.

full rationale

The model equations (3) and (4) reduce to algebraic consequences of the MIP definitions (Eqs. 1 and 2) under the explicit assumption that charge redistribution is negligible—a point the paper itself concedes in its summary ('becomes linear by definition'; 'the integral stays approximately constant'). The MASA derivation path is invoked but never executed ('Assuming that we would have computed MASAs for forward scattering' appears twice), making it a non-load-bearing detour rather than a circular self-citation chain. Self-citations to [4, 5, 6, 7, 12] by overlapping authors provide background context but are not load-bearing for the model equations, which follow from Eqs. 1–2 directly. Crucially, the paper provides genuine independent validation: DFT computations (Figs. 1–3) test the model against first-principles calculations that do not assume the model's premises, and the model uses only endpoint MIPs and lattice parameters as inputs. The agreement between model and DFT is a real empirical finding about whether the stated assumptions hold, not a tautology. Score 3 reflects that the model equations are near-definitional but the paper is transparent about this and provides independent DFT validation.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

No new entities are postulated. The model uses only existing physical quantities (MIP, volume, concentration) and standard DFT methodology.

free parameters (1)
  • None fitted = N/A
    The model contains no fitted parameters. Eq. 3 uses endpoint MIP values (from DFT or experiment) and Eq. 4 uses endpoint volumes. The comparison with DFT uses polynomial fits only for display, not as part of the model.
assumptions (4)
  • domain assumption Charge redistribution due to alloying, straining, and surface relaxation is significantly smaller than the difference between the crystal potential and isolated-atom potentials.
    Stated explicitly in the paragraph following Eq. 4: 'This simple approach of course only holds if the influence of any charge redistribution due to alloying, straining, surface relaxation etc. is significantly smaller than the difference to isolated atoms.' This is the load-bearing assumption for both Eq. 3 and Eq. 4.
  • ad hoc to paper Modified atomic scattering amplitudes (MASAs) for forward scattering would yield a linear combination rule for MIPs.
    Invoked in the derivation of Eq. 3: 'Assuming that we would have computed MASAs for forward scattering, expanding eq. (2) directly shows that the MIP of an alloyed material is a linear combination.' The MASA concept is never actually applied for forward scattering in this paper.
  • domain assumption The integral of the Coulomb potential over the unit cell stays approximately constant when the volume changes.
    Underlies Eq. 4: if the integral changes significantly with volume, the simple 1/Ω scaling fails. The authors note that the muffin-tin sphere contributions dominate (~94% for GaP), which supports this for small deformations.
  • domain assumption Vegard's law holds for lattice parameter interpolation in the alloy systems studied.
    Used for computing lattice parameters at intermediate compositions: 'only the change of lattice parameter according to Vegard's law and no strain was applied to the cells.'

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Cite this review

Pith. "Pith review of Modelling the mean inner potential of alloyed and strained materials." pith.science (2026). https://pith.science/paper/ZABGUAZC

@misc{pith2026260705948,
  author       = {Pith},
  title        = {Pith review of: Modelling the mean inner potential of alloyed and strained materials},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZABGUAZC}},
  note         = {Machine review of arXiv:2607.05948}
}
read the original abstract

In this publication, we study the influence of strain and alloying on the mean inner potential (MIP) using density functional theory (DFT) within an augmented plane waves plus local orbitals basis set. Two major effects have been identified allowing to model the influence of strain and alloying on the mean inner potential with a reasonable accuracy. First, alloying for constant volume results in a linear relationship between the MIP and the concentration. Second, the MIP scales with changes in volume as we already pointed out in an earlier publication (M. Schowalter, D. Lamoen, A. Rosenauer, P. Kruse, and D. Gerthsen, Appl. Phys. Lett. 85, 4938-4940 (2004)). Specifically, a linear relationship between MIP and concentration x was found for AlGaAs (nearly no change in lattice parameter), whereas InGaP and GeSi (volume changes with concentration x) exhibits a clear bowing. The bowing can be modeled by taking the rescaling of the MIP with the varying volume additionally into account. The rescaling could be also used to model the dependence of the MIP on strained binary cells and the density dependence of e.g. amorphous materials.

Figures

Figures reproduced from arXiv: 2607.05948 by the authors.

Figure 1
Figure 1. FIG. 1. The value of the MIP as function of the factor [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. shows the computed MIP as a function of concentration for unstrained a) AlxGa1−xAs, b) InxGa1−xP and c) GexSi1−x as blue circles as well as respective fits with a 2nd order polynomial (blue solid lines). For these computations only the change of lattice parameter according to Vegard’s law and no strain was applied to the cells. Green and red solid lines indicate the results of the proposed model only considering all… view at source ↗

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Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

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    (2) directly shows that the MIP of an alloyed materialV AxB1−x 0 is a linear combination of the MIPs of the unalloyed materialsV A 0 andV B 0 V AxB1−x 0 =xV A 0 + (1−x)V B 0 .(3)

    Assuming that we would have computed MASAs for forward scattering, expanding eq. (2) directly shows that the MIP of an alloyed materialV AxB1−x 0 is a linear combination of the MIPs of the unalloyed materialsV A 0 andV B 0 V AxB1−x 0 =xV A 0 + (1−x)V B 0 .(3)

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    Assuming that we would have computed MASAs for forward scattering, equation (2) suggests that the influence of any change in volume on the MIP could be calculated just by dividing by the respective volume. In other words, if the MIP V0(Ω) at a volume Ω of a system is known than the MIPV ′ 0(Ω′) of the system at a volume Ω ′ can be derived by V ′ 0(Ω′) = Ω...

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