Meyer-Neldel and anti-Meyer-Neldel rule in microcrystalline silicon and silicon carbide examined with Hall measurements

Oleksandr Astakhov; Reinhard Carius; Torsten Bronger

REVIEW 3 major objections 4 minor 28 references

Meyer-Neldel and anti-Meyer-Neldel rule in microcrystalline silicon and silicon carbide examined with Hall measurements

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A single microcrystalline silicon sample switches between Meyer–Neldel and anti-Meyer–Neldel behavior when its defect density changes, tying the rule to Fermi-level position.

desk verdict A useful experimental paper with a clever theoretical framing, but the headline 'first sample switching' rests on a by-eye split of six points and needs statistical backup before it can be believed. read the letter →

arxiv 1908.06516 v1 pith:EXGJHB7K submitted 2019-08-18 cond-mat.mtrl-sci

Torsten Bronger , Oleksandr Astakhov , Reinhard Carius This is my paper · ORCID

classification cond-mat.mtrl-sci

keywords Meyer-Neldelruleanti-Meyer-NeldelmicrocrystallinesiliconcarbideHalleffectstatisticalshiftcarrierconcentrationmobility

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 examines the Meyer–Neldel rule (MNR) and its inverse, anti-MNR, in hydrogenated microcrystalline silicon and silicon carbide, using temperature-dependent Hall measurements to separate conductivity into carrier concentration and mobility. Its central claim is that the same 15 ppm microcrystalline silicon sample switches from anti-MNR to normal MNR when its defect density is changed by electron bombardment and stepwise annealing, the first such transition achieved by changing material properties rather than gate voltage. The authors argue this supports the statistical-shift picture, in which the Fermi-level position, not sample-to-sample variation, determines MNR behavior. They also report anti-MNR in the carrier concentration and conductivity of silicon carbide with parameters close to silicon, and they derive a power-law relation between room-temperature mobility and carrier concentration from an observed linear proportionality between their activation energies.

What carries the argument

The central object is the Meyer–Neldel rule itself, the observation that Arrhenius prefactors grow exponentially with activation energy so that extrapolated Arrhenius lines meet at one point, together with its negative-slope counterpart anti-MNR. The theoretical machinery is the statistical-shift model, which predicts MNR for carrier concentration because the Fermi level moves with temperature. The paper shows this model alone does not give MNR in conductivity; the mobility must meet one of three conditions. The load-bearing condition in this work is the empirical proportionality $E_n^a = a\,E_\mu^a + b$, observed with $a \approx 1.25$ and $b \approx -18$ meV in the anti-MNR domain of both $\mu$c-Si:H and $\mu$c-SiC:H. Substituting this relation into the Arrhenius forms for $n$ and $\mu$ yields the room-temperature power law $\mu_r(n_r) = \mu'_{00} (n_r/n_{00})^\kappa$ with $\kappa = E^n_{MN}(E^\mu_{MN} - kT_r)/(a\,E^\mu_{MN}(E^n_{MN} - kT_r))$.

What would settle it

Measure Hall transport on a sample whose defect density is stepped finely across the anti-MNR-to-MNR boundary: if the exponent $\kappa$ in $\mu_r(n_r)$ changes with the transition, or if $E_n^a$ versus $E_\mu^a$ stops lying on the $1.25$ slope line, then the derived power law and the claim that this proportionality governs the anti-MNR domain are wrong.

Watch

Extended reading notes

Core claim

Using Hall measurements on seven microcrystalline silicon layers, one amorphous layer, two irradiated-and-annealed silicon samples, and five silicon carbide layers, the paper establishes three results. First, all three transport quantities—conductivity, carrier concentration, and mobility—show anti-Meyer–Neldel behavior in lightly doped microcrystalline silicon, and silicon carbide shows anti-MNR in conductivity and carrier concentration with Meyer–Neldel prefactors and energies close to silicon's. Second, the 15 ppm sample Ma-15 moves from the anti-MNR branch to the normal MNR branch as annealing removes bombardment-induced defects; the authors state this is the first report of such a transition by changing material properties, and take it as evidence that Fermi-level position controls MNR. Third, in the anti-MNR domain, the activation energies for carrier concentration and mobility obey a common linear relation $E_n^a = 1.25\,E_\mu^a - 18$ meV for both materials; combined with anti-MNR this yields a power law $\mu_r(n_r) \propto n_r^\kappa$, with $\kappa$ between 0.5 and 1.5, which the annealed-sample data reproduce with exponents around 0.48 and 0.61.

Load-bearing premise

All the quantitative conclusions rest on treating the measured transport as a single channel with well-defined activation energies, even though the temperature plots bend and the paper says the cause of the bending is speculative.

Editorial extensions

If this is right

A fixed sample can be moved across the Meyer–Neldel/anti-Meyer–Neldel boundary solely by changing its defect density, so Fermi-level position, not sample identity, is what selects the regime.
Observing MNR in conductivity is not automatic once carrier concentration follows the statistical shift; the mobility activation energy must satisfy one of three stated conditions, and in these data it does so through the linear $E_n^a$–$E_\mu^a$ relation.
Silicon carbide joins silicon with nearly the same anti-MNR parameters, strengthening the case for a common Meyer–Neldel behavior across disordered semiconductors.
The derived power law $\mu_r \propto n_r^\kappa$ with $\kappa$ between 0.5 and 1.5 gives a compact way to predict room-temperature mobility from carrier concentration in the anti-MNR regime.
The unresolved item the paper leaves open is a microscopic explanation for why $E_n^a$ and $E_\mu^a$ are proportional in the first place.

Reading between the lines

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

If the proportionality $E_n^a = 1.25\,E_\mu^a - 18$ meV holds beyond these two material families, then the ratio between carrier-concentration and conductivity Meyer–Neldel energies (about 1.7 here) should be derivable from that line alone; a future analysis could check that consistency.
The same-sample switch suggests a device-relevant handle: irradiation and low-temperature annealing can toggle a film's transport regime without changing its chemistry.
A sharper test than the paper's would fix the exact defect densities and measure $\kappa$ continuously across the anti-MNR-to-MNR boundary; if $\kappa$ is constant within the anti-MNR branch, the power-law picture is robust, and if it drifts, the boundary is not a clean phase-like division.
Because the paper lists barrier-height distributions and differential mobility as alternative causes of the Arrhenius curvature, a natural next step is to check whether either mechanism also predicts the measured $E_n^a$–$E_\mu^a$ slope; if one does, it would unify the curvature and the power law.

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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

The paper is more interesting than its low-key framing suggests, but the one claim the authors push hardest -- that a single sample switches between anti-MNR and normal MNR when annealed -- is also the weakest supported.

What's actually new: first report of anti-MNR in μc-SiC:H, with σ00 and E_MN matching the silicon values, which strengthens the universality observation. And the proportionality E_n = 1.25 E_mu − 18 meV across very different samples is a genuinely useful empirical result, especially if it holds up under more careful fitting. The theoretical section is a good piece of clarity: it spells out that MNR in n does not guarantee MNR in σ, and the three conditions for σ MNR are worth having. The derivation of the mu(n) power law from the proportionality is algebraically fine, though the authors overstate it as a prediction when it's really a rearrangement of fitted parameters.

The soft spot is the switching claim. In Fig. 5, six points of the 15 ppm sample are split into three left and three right by eye, with no fit and no goodness-of-fit test for the right group. The text concedes those points have larger error bars. If those three points are statistically consistent with the anti-MNR line, the transition evaporates. Given the Arrhenius curves are convex, the activation energies are local slopes at room temperature, so the grouping could shift with the fitting window. The authors need a simple chi-square or t-test on the split, or a combined fit with a breakpoint, to make the claim stick.

Minor issues: Fig. 8 fit has no reported uncertainties; the text says the numbers 'must be taken with care' but does not propagate them into kappa. And the power-law exponent is fitted again in Fig. 9 after being derived, which is circular -- report it as a consistency check.

Citation pattern is normal; prior work is cited appropriately.

Bottom line: this deserves peer review, not desk rejection. The experimental data on n and mu in these materials is rare and presumably hard to get. The switching claim needs real statistical support, and the theoretical claims need to be scaled back to consistency checks. A serious referee would ask for those changes, and the paper would be stronger for it.

Referee Report

3 major / 4 minor

Summary. The paper reports temperature-dependent Hall measurements of conductivity, carrier concentration, and mobility for lightly phosphorus-doped microcrystalline silicon (μc-Si:H) and hydrogenated silicon carbide (μc-SiC:H), including a series of electron-irradiated μc-Si:H samples measured after step-wise annealing. The authors analyze the data in Meyer-Neldel rule (MNR) plots and report anti-MNR behavior for most samples, the first observation of anti-MNR in μc-SiC:H, and a claim that a single μc-Si:H sample (Ma-15) switches from anti-MNR to normal MNR as its defect density is changed by annealing. The theoretical part of the paper shows that MNR in the carrier concentration does not by itself imply MNR in conductivity unless the mobility satisfies one of three conditions, identifies the proportionality E_n^a = a E_μ^a + b as the operative condition, and derives from it a power-law relation between room-temperature mobility and carrier concentration, μ_r = μ'_00 (n_r/n_00)^κ.

Significance. If the empirical claims hold, the paper provides a valuable experimental dataset that extends the MNR phenomenology: the first reported anti-MNR in μc-SiC:H, a systematic comparison of MNR in σ, n, and μ for the same samples, and a claimed single-sample transition between anti-MNR and MNR that would support the Fermi-level position as the controlling parameter. The theoretical derivation connecting the linear E_n^a-E_μ^a relation to a μ(n) power law is a useful formal step, even though its practical value depends on the independence of the fitted parameters. The paper also gives credit by explicitly flagging the speculative origin of the Arrhenius curvature and the large uncertainties in the mobility data, which is appropriate for a study of this type.

major comments (3)

[IV A, Fig. 5] The central claim that sample Ma-15 switches from anti-MNR to normal MNR is based on a visual dichotomy of six data points, three assigned to the anti-MNR line and three to a dashed 'guide to the eye' for normal MNR. No fit is reported for the right-hand group, no goodness-of-fit test is compared with the anti-MNR line, and the text states that these points have 'larger error estimates.' The authors should quantify whether the three right-hand points are statistically incompatible with the anti-MNR line fitted from the annealed samples, including the effects of the large error bars and of the choice of Arrhenius fit window, before the switching claim can be accepted.
[V B, Eq. (7) and Fig. 9] The power-law exponent κ in Eq. (7) is computed from E^n_MN, E^μ_MN, a, and b, all of which are fitted from the same Hall data used to produce Fig. 9. The subsequent fit of κ in Fig. 9 (values 0.48 and 0.61) is therefore not an independent test of the derived relation; it only checks that the direct fit lies in the broad range allowed by the large uncertainties. The authors should either provide a proper uncertainty propagation for κ and state whether the directly fitted exponents are consistent with the derived values under that propagation, or clearly label the power law as a consistency check rather than a prediction.
[III and Figs. 5-8] The MNR coordinates (activation energy and prefactor) are extracted from Arrhenius curves that are explicitly convex rather than straight, and the text concedes that the cause of the curvature is speculative. Because each MNR point depends on the local slope and intercept at room temperature, the classification of samples into MNR and anti-MNR regimes, including the claimed Ma-15 switching, is sensitive to the fitting procedure. The manuscript should specify the exact fitting window and method, and ideally show how the MNR plot changes when the window is varied within the measurement range, so that the reader can assess whether the reported separation is robust.

minor comments (4)

[IV A] The text states that the demarcation between MNR and anti-MNR lies at 'approx. 120 eV'; this should presumably be 120 meV.
[II B and Fig. 9] There are typographical errors: 'degration' in Section II B, 'poylnomial' in the caption of Fig. 9, and 'suprising' in Section IV B. The sample identifiers '05B-054' and '06B-276' in Section V B are not defined in the paper.
[Appendix A] The appendix title says 'at root temperature'; this should be 'at room temperature.'
[V A, Eq. (4)] The notation 'terms without Ea’s' is imprecise; the authors should state explicitly which constants (such as n_00 and μ_00) are absorbed into the y-intercept, to make the derivation easier to follow.

Circularity Check

1 steps flagged · score 6.0 of 10

The derived µ(n) power law is a rearrangement of fitted MNR parameters and the fitted E_n–E_µ relation; the experimental MNR and switching observations themselves are not circular.

fitted input called prediction [Section V B, Eqs. (5)–(7), Fig. 9, and Appendix A]
"This polynomial relation can be derived from the presence of anti-MNR and the linear dependence of the activation energies of n and µ: ... κ = ... with Tr being room temperature, a and b as defined in the previous section, and κ ranging between 0.5 and 1.5, in accordance with the large uncertainties in EnMN and EµMN. ... Fig. 9 ... The given values for κ represent the exponent of the polynomial fits ... and refer to Eq. 7."

The power-law exponent κ in Eq. 7 is expressed in terms of E^n_MN, E^µ_MN, a, and b. E^n_MN and E^µ_MN are fitted from the same annealed-sample Hall data (Table III, Figs. 6 and 7), and a and b are fitted from the same samples' E^n_a versus E^µ_a relation (Fig. 8, Eq. A3). The paper then presents Fig. 9, where κ is fitted again from the same room-temperature µ–n data, and accepts agreement only within the broad 0.5–1.5 range. Since the power law is an algebraic consequence of the Arrhenius/MNR forms and the fitted linear relation, any data set satisfying those fitted relations will by construction exhibit a power law with κ in that range. The 'derivation' therefore reduces to the fitted inputs rather than providing an independent prediction.

full rationale

The paper's empirical findings — anti-MNR for σ, n, and µ, the first anti-MNR observation in µc-SiC:H, and the claimed same-sample MNR/anti-MNR switching — are direct descriptions of Hall measurements and are not circular in themselves; they stand or fall on data quality and statistical analysis. The circularity lies in the theoretical section. The claimed derivation of the µ(n) power law (Eq. 5) uses parameters that are all fitted from the same dataset: the Meyer-Neldel energies from the anti-MNR fits of the annealed samples (Table III) and the slope/intercept a and b of the E^n_a–E^µ_a proportionality (Fig. 8). Inserting these fitted values into Eq. 7 gives κ, and Fig. 9 then fits κ from the same room-temperature n–µ points, comparing only against the loose 0.5–1.5 interval. This is not an independent test but a consistency check, and the 'prediction' is forced by the construction of the equations. I did not find other load-bearing circular steps: the statistical-shift assumption is a stated modeling choice, and the self-citations (e.g., Ref. 13 for previously published Hall data) are not relied on as the sole justification for any central result. The score of 6 reflects partial circularity affecting the theoretical derivation while the core experimental observations retain independent content.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central experimental observations rest on standard but unverified transport assumptions. The theoretical derivation then adds fitted parameters (MNR energies and the E_n-E_mu line) whose values come from the same Hall data that the power-law relation is checked against. No new physical entities are introduced.

free parameters (6)

E^sigma_MN (anti-MNR energy for conductivity) = -17 +/- 6 meV
Fitted from the MNR plot of conductivity for annealed samples (Table III); used to support the anti-MNR assignment and to compare with the n and mu results.
E^n_MN (anti-MNR energy for carrier concentration) = -29 +/- 3 meV
Fitted from the MNR plot of carrier concentration (Table III); enters Eq. 7 through the derived power-law exponent.
a (slope of E_n vs E_mu) = 1.25
Linear fit in Fig. 8; central input to condition 3 that makes MNR in sigma possible and to the mu(n) power-law derivation.
b (intercept of E_n vs E_mu) = -18 meV
Same linear fit in Fig. 8; used in Eq. 6 for the prefactor mu'_00.
E^mu_MN (anti-MNR energy for mobility) = not reliably determined
The paper states uncertainties are too large for quantitative evaluation, yet Eq. 7 depends on E^mu_MN, which is why the derived kappa range is broad (0.5-1.5).
demarcation activation energy between MNR and anti-MNR = approx. 120 meV
Empirically read from Fig. 5; used to classify samples into MNR and anti-MNR regimes, but not central to the theoretical derivation.

assumptions (5)

domain assumption The statistical shift model describes the temperature dependence of carrier concentration n in these samples.
Invoked in Section III to explain convex Arrhenius curvature and in Section V A as the source of MNR in n; no independent verification is provided for these specific materials.
domain assumption Each sample is characterized by single activation energies E_n and E_mu over the fitted temperature window.
Used to extract slopes and intercepts for all MNR plots; the curves are convex, so the results depend on the choice of room temperature as the reference point.
ad hoc to paper The relation E_n = a E_mu + b is linear and holds for all anti-MNR samples.
Postulated from the fit in Fig. 8 after MNR samples are excluded; no theoretical derivation or detailed uncertainty analysis is given.
domain assumption Hall measurements yield the true free-carrier concentration n and mobility mu.
Standard for homogeneous semiconductors; in mixed-phase microcrystalline Si and SiC, Hall scattering factors and inhomogeneities can bias results. The paper itself notes sign reversal in the amorphous sample and excludes it.
domain assumption The mobility has a nonzero activation energy E_mu, so mobility-edge theory with mu ~ T^-1 does not apply.
The paper argues from Fig. 8 that E_mu is nonzero and builds the MNR condition on that basis; it is an empirical assumption rather than a derived result.

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Pith. "Pith review of Meyer-Neldel and anti-Meyer-Neldel rule in microcrystalline silicon and silicon carbide examined with Hall measurements." pith.science (2026). https://pith.science/paper/EXGJHB7K

@misc{pith2026190806516,
  author       = {Pith},
  title        = {Pith review of: Meyer-Neldel and anti-Meyer-Neldel rule in microcrystalline silicon and silicon carbide examined with Hall measurements},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EXGJHB7K}},
  note         = {Machine review of arXiv:1908.06516}
}

abstract

We study the electronic transport in lightly phosphorus-doped hydrogenated microcrystalline silicon ($\mu$c-Si:H) and nominally undoped hydrogenated silicon carbide ($\mu$c-SiC:H) by temperature-dependent Hall measurements. The material properties cover different crystallinities and doping concentrations. For$\mu$c-Si:H samples, the carrier concentration is altered by electron bombardment and subsequent step-wise annealing of defects. We describe the behavior of conductivity, mobility, and carrier concentration in terms of the Meyer-Neldel rule (MNR) and anti-MNR. We present the first sample switching between them. A theoretical examination leverages the anti-MNR to describe electronic room temperature properties, and it expands the statistical shift model.

Figures

Figures reproduced from arXiv: 1908.06516 by the authors.

**Figure 2.** FIG. 2. Temperature-dependent conductivity for the degraded sample Ma-150, for different annealing steps. [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** FIG. 3. Temperature-dependent carrier concentration for the degraded sample Ma-150, for different annealing steps. [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: FIG. 4. Temperature-dependent mobility for the degraded sample Ma-150, for different annealing steps. [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5. MNR plot of conductivity for various measurement series. Data of the annealed samples (green and red) has error [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 6.** Figure 6: FIG. 6. MNR plot of carrier concentration for various measurement series. The yellow solid line is a linear fit considering only [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

**Figure 7.** Figure 7: FIG. 7. MNR plot of mobility for various measurement series. Data of the annealed samples (green and red) has error bars. [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]

**Figure 8.** Figure 8: FIG. 8. Activation energy of carrier concentration versus activation energy of mobility of various measurement series. Only [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

**Figure 9.** Figure 9: FIG. 9. The mobility versus the carrier concentration of the annealed samples, both measured at room temperature. The given [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

e.,µ is temperature-independent)

Eµ a = 0 (i. e.,µ is temperature-independent)

work page

[2] [2]

W. Fuhs, P. Kanschat, and K. Lips, Journal of Vacuum Science & Technology B: Microelectronics and Nanometer Structures 18, 17921795 (2000)

work page 2000

[3] [3]

The respective MNR energies for σ are:

En a =a·Eµ a +b Here, a and b are parameters of the linear dependence. The respective MNR energies for σ are:

work page

[4] [4]

the conductivity in a-Si:H exhibits a MNR because the statistical shift causes the carrier density to follow a MNR as well as the mobility

Eσ MN = (a + 1) En MNEµ MN aEµ MN +En MN Thus, the statistical Fermi level shift is not suﬃcient for the occurrence of some form of MNR in σ. Additionally, the mobility must fulﬁll at least one of the three conditions. Generally, examinations of the MNR in disordered semiconductors implicitly assume (1), i. e. a very weak temper- ature dependence of mobil...

work page

[5] [5]

W. v. Meyer and H. Neldel, Z. tech. Phys 18, 588 (1937)

1937

[6] [6]

Kondo, Y

M. Kondo, Y. Chida, and A. Matsuda, Journal of Non-Crystalline Solids 198, 178 (1996)

work page 1996

[7] [7]

Thomas and H

P. Thomas and H. Overhof, in Defect and Diﬀusion Forum , Vol. 192 (Trans Tech Publ, 2001) p. 114

work page 2001

[8] [8]

S. K. Ram, S. Kumar, and P. Roca i Cabarrocas, Journal of Non-Crystalline Solids 354, 22632267 (2008)

work page 2008