Non-localized states and high hole mobility in amorphous germanium

Anders Hallen; Jennifer Wong-Leung; Jim S. Williams; Lachlan A. Smillie; Maria G. Grimaldi; Tuan T. Tran

REVIEW 3 major objections 4 minor 31 references

Non-localized states and high hole mobility in amorphous germanium

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

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Pith's one-line read Ion-implanted amorphous germanium conducts holes at about 100 cm²/(V·s), and the paper reads this as direct evidence of a preserved, non-localized valence band.

desk verdict Ion-implanted a-Ge shows a remarkably high Hall mobility, but the substrate-leakage control is missing and the 'direct evidence' language outruns the data. read the letter →

arxiv 1908.08246 v1 pith:CRLRYP3E submitted 2019-08-22 cond-mat.mtrl-sci

Tuan T. Tran , Jennifer Wong-Leung , Lachlan A. Smillie , Anders Hallen , Maria G. Grimaldi , Jim S. Williams This is my paper · ORCID

classification cond-mat.mtrl-sci

keywords amorphousgermaniumionimplantationHalleffectholemobilitynon-localizedstatesvalencebandsemiconductorhoppingtransport

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

This paper tries to establish that amorphous germanium is not inherently a poor conductor. It reports that a-Ge made by implanting Ge ions into a Ge-on-Si wafer is dense, homogeneous, and free of voids, and that Hall measurements show p-type conduction with a hole mobility around 100 cm²/(V·s), four orders of magnitude above conventionally deposited a-Ge. The resistivity has two well-defined regimes, extrinsic and intrinsic, resembling crystalline Ge rather than the gradual rise typical of evaporated films. The authors conclude that the valence band of dense a-Ge is largely preserved, with extended non-localized states, and that the low mobilities reported for deposited films came from density fluctuations and defects rather than from amorphous disorder itself. If true, this would overturn the default assumption that amorphous covalent semiconductors conduct only by hopping.

What carries the argument

The load-bearing object is the dense amorphous germanium film itself: implanting 2.5 MeV 74Ge+ ions at 1×$10^{15}$ $cm^{-2}$ into a Ge-on-Si substrate held at liquid-nitrogen temperature produces full amorphization (dpa > 2) and a void-free layer whose mass density is within 98.5% of crystalline Ge, sitting on an amorphized Si layer that is supposed to be 4-6 orders of magnitude more resistive. The argument is carried by van der Pauw Hall measurements from 30 K to 360 K, using the single-carrier identities p = α/(R_H e) and μ = R_H/ρ, plus a 1966 optical analysis of a-Ge showing valence-band transitions described by the same formula set as crystalline Ge. What connects density to transport is the proposal that density fluctuations in evaporated films create potential barriers that localize carriers, whereas the homogeneous implanted network preserves extended valence-band states.

What would settle it

Etch away the a-Ge layer and measure the exposed amorphous-silicon/crystalline-silicon stack under the same van der Pauw geometry: if its resistance or Hall response is within four orders of magnitude of the intact sample at any temperature, the reported mobility is contaminated by substrate conduction. Alternatively, amorphize a Ge film on an insulating substrate and check that the ~100 cm²/(V·s) mobility and the extrinsic-to-intrinsic resistivity transition survive.

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Extended reading notes

Core claim

The central claim is that ion-implanted amorphous germanium is a p-type semiconductor with a hole mobility of order 100 cm²/(V·s), a carrier density near $10^{17}$ $cm^{-3}$, and a resistivity that switches sharply from extrinsic to intrinsic conduction just as crystalline Ge does. These electrical signatures are presented as direct evidence that the valence band of dense a-Ge retains a largely intact band structure and that its hole wave functions are extended rather than localized, an idea previously inferred from optical absorption alone. The paper attributes the four-order-of-magnitude gap with earlier evaporated films to mass-density deficits and impurity-related potential barriers in deposited material, not to an intrinsic property of the amorphous network.

Load-bearing premise

The measurement assumes that the Hall voltage and resistance come entirely from the amorphous germanium layer, not from the amorphized silicon interlayer or the crystalline silicon substrate, which the paper estimates to be 4-6 orders of magnitude more resistive but does not verify with a direct electrical control.

Editorial extensions

If this is right

Implanted a-Ge shows a sharp extrinsic-to-intrinsic resistivity transition, so doping and impurity control can work in principle much as they do in crystalline Ge.
Deposited films with 72-95% of crystalline density are not reliable stand-ins for intrinsic a-Ge; their low mobility reflects growth-related barriers, not the amorphous network itself.
Implanted a-Si remains a hopping conductor with resistivity 4-6 orders of magnitude above implanted a-Ge, so non-localized transport in the amorphous phase is not universal across covalent semiconductors.
A high-mobility amorphous germanium opens device applications in thin-film electronics, solar cells, and flexible devices that had been considered closed to amorphous semiconductors.

Reading between the lines

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

The paper does not directly measure the resistance of the a-Si/c-Si stack after removing the a-Ge layer; a control measurement of that kind is the cleanest way to rule out substrate leakage as the source of the 100 cm²/(V·s) mobility.
If the density explanation is correct, deposition methods that approach crystalline density and exclude impurities should reproduce mobilities within an order of magnitude of 100 cm²/(V·s), turning the proposal into a predictive design rule for amorphous films.
The claim implies a concrete electronic-structure prediction: a continuous random network of Ge at near-crystalline density should have a delocalized valence-band edge in atomistic calculations, so a numerical search for band-edge localization could settle the conflict with hopping-based 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

The paper reports a ~100 cm²/Vs Hall mobility in dense ion-implanted a-Ge, four orders of magnitude above evaporated films, and frames it as direct evidence for non-localized valence-band states. If the transport measurement is clean, that is a big deal and would reconcile Tauc's optical picture with electrical data. The sample preparation looks careful: the TEM is convincing, the c-Ge reference matches literature, and the contrast with evaporated a-Ge is stark. That part is solid.

The problem is the substrate. The a-Ge sits on a 1.1 µm a-Si layer that is supposed to insulate it from the c-Si substrate. The paper argues a-Si is negligible because its resistivity is 4–6 orders higher, but that comparison is for lateral sheet resistance. Vertical leakage through the a-Si into the conducting substrate could be tiny: at 360 K the a-Si resistivity is ~10² Ω cm, giving a perpendicular resistance of order 1 Ω for a square-centimeter sample. That can short the a-Ge. The earlier ion-implanted study cited as inconclusive was indeed undermined by the substrate, and the authors' remedy is exactly the layer that needs an electrical control. Etching away the a-Ge and re-measuring, or measuring a plain a-Si sample on c-Si, would settle it. Without that, the 100 cm²/Vs figure could be an artifact of the substrate path.

Other soft spots are minor by comparison. There are no error bars or raw data for the key Hall and resistivity points. The step from high Hall mobility to 'largely preserved band structure' is an interpretation, not a measurement. The Hall scattering factor is dismissed as nearly unity, which is standard but not rigorous at the claimed level.

The citation pattern looks fine, and the authors know the prior art. The paper is clearly written and makes a testable claim. It deserves a careful referee, but the referee should send it back for a substrate-isolation control before the result can be taken at face value.

Referee Report

3 major / 4 minor

Summary. This manuscript reports temperature-dependent Hall effect and resistivity measurements on amorphous germanium (a-Ge) layers created by 2.5 MeV 74Ge+ self-ion implantation into Ge-on-Si substrates. The authors find p-type conductivity with a carrier density of order 10^17 cm^-3 and a hole mobility of order 100 cm^2/(Vs) in the extrinsic region (T < 275 K), and a resistivity curve with distinct extrinsic and intrinsic regions that they describe as resembling crystalline Ge. They interpret these results as direct evidence for largely preserved valence-band structure and non-localized states in dense a-Ge, supporting the earlier optical proposal of Tauc et al. The paper argues that the underlying 1.1 um amorphous Si (a-Si) layer is sufficiently resistive that its contribution to the electrical measurements is negligible.

Significance. If the measured transport properties are intrinsic to the a-Ge layer, the result would be substantial: it would demonstrate extended-state hole transport in a covalent amorphous semiconductor with a mobility four orders of magnitude higher than typical evaporated a-Ge, reconciling electrical transport with earlier optical data and opening possible device applications. The paper also contains useful experimental details, including liquid-nitrogen-temperature implantation to avoid porosity, TEM confirmation of a void-free 0.91 um a-Ge layer, and direct comparison with crystalline Ge and evaporated a-Ge. However, the central claim rests on excluding parallel conduction through the a-Si interlayer and the c-Si substrate, and that exclusion is not convincingly established in the manuscript.

major comments (3)

[Sample structure and electrical characterization (pp. 4-7)] The argument that the a-Si layer contributes negligibly to the electrical measurements is based on comparing resistivities, not resistances. For the vertical leakage path from the a-Ge layer through the 1.1 um a-Si layer into the conductive c-Si substrate, the relevant resistance is rho_aSi*t_aSi/A. Taking rho_aSi ~ 10^4 Ohm-cm at 300 K and a contact area of order 10^-3 cm^2 gives a vertical resistance of about 10^3 Ohm, which is comparable to the ~5x10^3 Ohm Van der Pauw resistance expected for the 0.91 um a-Ge layer (rho ~ 0.1 Ohm-cm). At 360 K, where rho_aSi drops to ~10^2 Ohm-cm, this shunt resistance falls to ~10 Ohm, making a parallel path through the substrate potentially dominant. The manuscript does not provide an electrical control, such as etching away the a-Ge layer and re-measuring the remaining stack, or a quantitative analysis of the full current path including contact geometry. Without such a control, the extracted mobility of ~100 cm^2/(Vs) may be an artifact of parallel conduction through the a-Si/c-Si substrate rather than a property of the a-Ge layer.
[Abstract and Conclusion] The statement that the electrical results are 'direct evidence for a largely-preserved band structure' overreaches the measurements. The paper presents resistivity and Hall data, but no band-structure-resolved probe such as optical absorption, photoemission, or a quantitative comparison of the intrinsic activation energy with that of crystalline Ge. The earlier optical work of Tauc et al. is an external prior and is not established by the present transport data. The conclusion should be tempered to state that the transport data are consistent with extended valence-band states in dense a-Ge, rather than claiming direct proof of a preserved band structure.
[Carrier density analysis (p. 7)] The paper assumes a Hall scattering factor alpha = 1 in Eq. (1). While alpha is often close to unity for extended-state transport, the manuscript does not justify this assumption for amorphous germanium, where Hall effect anomalies are known. If the measured signal contains contributions from the substrate or if the transport mechanism is not simple single-carrier extended-state conduction, the extracted carrier density and mobility could be systematically in error. At minimum, the sensitivity of the extracted mobility to realistic values of alpha (e.g., 0.5-2) should be discussed.

minor comments (4)

[Fig. 2 caption and accompanying text (p. 6)] The sign convention for the triangles is inconsistent. The text immediately before Fig. 2 states that empty triangles indicate negative R_H and filled triangles indicate positive R_H, whereas the Fig. 2 caption says R_H is positive for empty triangles and negative (plotted positive) for filled triangles. Please correct one of the descriptions.
[References] Reference [22] is incomplete: 'J. W. M. M. Nastasi, (2006)' lacks a title and publisher information.
[General experimental reporting] No error bars or multiple-sample statistics are provided. Given that the observed mobility is four orders of magnitude above literature values for evaporated a-Ge, reporting data from at least two or three independently prepared samples would strengthen confidence in the reproducibility of the result.
[Section 'The starting samples' (p. 4)] The phrase 'resistance ratio of the a-Si layer over the a-Ge layer' is ambiguous because resistance depends on the geometry of the current path. The later comparison in Section 3 (Fig. 2a) is of resistivities, not of the resistances encountered in the Van der Pauw measurement. Please either specify the geometry used for the ratio or remove the phrase.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the Hall mobility and carrier density are direct measurements via textbook formulas, with no fitted parameter or self-referential construction.

full rationale

The central derivation chain is self-contained. The paper measures the Hall coefficient R_H and resistivity rho of the implanted a-Ge layer and computes carrier density as p = alpha/(R_H e) and mobility as mu = R_H/rho, with alpha taken close to unity by standard convention. No parameter is tuned to produce the ~100 cm^2/(Vs) mobility, and no equation in the paper contains the target mobility as an input. The c-Ge control data are validated against independent literature values (Morin et al.), and the deposited a-Ge data are compared as an external benchmark. Tauc et al. is cited as an external optical prior proposal used for interpretation after the electrical measurement, not as a derivation input. The self-citations that appear (e.g., Refs. [14], [18], [21]) are used only for motivational or empirical support, such as the dpa threshold for amorphization or porosity effects, and are not load-bearing for the electrical extraction; Ref. [14] is explicitly described as inconclusive due to substrate contribution. The a-Si insulation argument is an empirical resistivity comparison rather than a reduction to the paper's own conclusion; whether the geometrical shunt is fully excluded is an experimental validity concern, not a circularity. Therefore no step reduces by construction to its own inputs.

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

The central claim rests on three unverified assumptions: the substrate is electrically isolated, the implanted film is fully amorphous and dense, and single-carrier Hall analysis applies. The mobility number itself is not fitted from a model; it is extracted from measured R_H and rho using textbook formulas, so the free-parameter count is low. The main interpretive burden is identifying high mobility with non-localized band states, which is a physical inference, not a derivation.

free parameters (1)

Hall scattering factor alpha = 1 (assumed)
In p = alpha/(R_H e), alpha is set to unity; the authors state it introduces only small error, but it is an unmeasured material-dependent factor that scales carrier density and mobility.

assumptions (4)

domain assumption The amorphized silicon interlayer is sufficiently insulating that the crystalline Si substrate does not contribute to Hall/resistivity measurements.
Invoked near the sample structure description to justify interpreting the multilayer measurement as pure a-Ge; no direct leakage test is provided.
domain assumption Ion implantation with dpa > 2 produces fully amorphous, non-porous Ge with density close to crystalline Ge.
Based on SRIM and prior studies [17,18], not verified in this paper by density measurement; TEM shows amorphous diffraction but not quantitative density.
domain assumption Single-carrier extrinsic conduction holds below 275 K, so p = alpha/(R_H e) and mu = R_H/rho apply.
The paper infers a single carrier from nearly temperature-independent resistivity; the Hall scattering factor alpha is set to 1.
domain assumption High Hall mobility in an amorphous film implies extended, non-localized valence-band states and a largely preserved band structure.
This is the interpretive step from transport data to band-structure conclusion; it is plausible but not directly proven by the electrical measurement.

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Cite this review

Pith. "Pith review of Non-localized states and high hole mobility in amorphous germanium." pith.science (2026). https://pith.science/paper/CRLRYP3E

@misc{pith2026190808246,
  author       = {Pith},
  title        = {Pith review of: Non-localized states and high hole mobility in amorphous germanium},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CRLRYP3E}},
  note         = {Machine review of arXiv:1908.08246}
}

read the original abstract

Covalent amorphous semiconductors, such as amorphous silicon (a-Si) and germanium (a-Ge), are commonly believed to have localized electronic states at the top of the valence band and the bottom of the conduction band. Electrical conductivity is thought to be by the hopping mechanism through localized states. The carrier mobility of these materials is usually very low, in the order of ~10^-3 - 10^-2 cm^2/(Vs) at room temperature. In this study, we present the Hall effect characterization of a-Ge prepared by self-ion implantation of Ge ions. The a-Ge prepared by this method is highly homogenous and has a mass density within 98.5% of the crystalline Ge. The material exhibits an exceptionally high electrical conductivity and carrier mobility (~100 cm^2/(Vs)) for an amorphous semiconductor. The temperature-dependent resistivity of the material is very-well defined with two distinctive regions, extrinsic and intrinsic conductivity, as in crystalline Ge. These results are direct evidence for a largely-preserved band structure and non-localized states of the valence band in a-Ge, as proposed by Tauc et al. from optical characterization alone. This finding is not only significant for the understanding of electrical conductivity in covalent disordered semiconductors, but the exceptionally high mobility we have observed in amorphous Ge opens up device applications not previously considered for amorphous semiconductors.

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]

Madan and M

A. Madan and M. P. Shaw, The physics and applications of amorphous semiconductors (Elsevier, 2012)

work page 2012

[2] [2]

A. H. Clark, Journal of Non-Crystalline Solids 2, 52 (1970)

work page 1970

[3] [3]

A. I. Gubanov, Quantum electron theory of amorphous conductors (Springer Science & Business Media, 1965)

work page 1965

[4] [4]

E. A. Davis and N. F. Mott, Philosophical Magazine 22, 0903 (1970)

work page 1970

[5] [5]

Tauc, Science 158, 1543 (1967)

J. Tauc, Science 158, 1543 (1967)

work page 1967

[6] [6]

Beyer and J

W. Beyer and J. Stuke, physica status solidi (a) 30, 511 (1975)

work page 1975

[7] [7]

P. A. Walley and A. K. Jonscher, Thin Solid Films 1, 367 (1968)

work page 1968

[8] [8]

A. H. Clark, Physical Review 154, 750 (1967)

work page 1967