Hot-Carrier Distribution Spectroscopy by Transconductance in Two-Dimensional Field-Effect Transistors

Katsunori Wakabayashi

REVIEW 4 major objections 4 minor 39 references

Hot-Carrier Distribution Spectroscopy by Transconductance in Two-Dimensional Field-Effect Transistors

T0 review · 4 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read The transconductance of a 2D FET is a spectral transform of the carrier distribution, not a density meter — and its shape-driven term carries an anomalous peak that reads out E0, σ, nc.

desk verdict A clean, testable proposal for reading hot-carrier distribution shape from transconductance, but the 'spectroscopy' claims outrun what the model can actually constrain. read the letter →

arxiv 2607.15578 v1 pith:4CRIKGQM submitted 2026-07-17 cond-mat.mes-hall cond-mat.mtrl-sci

Katsunori Wakabayashi This is my paper · ORCID

classification cond-mat.mes-hallcond-mat.mtrl-sci

keywords 2Dfield-effecttransistorstransconductancespectroscopyhot-carrierdistributionspectralcurrentkernelnonequilibriumtransportall-electricalcarrierrelaxationtimeMoS2

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 argues that the transconductance of a two-dimensional field-effect transistor is fundamentally a spectral probe of the carrier distribution, not merely a carrier-density meter: g_m weights the spectral current j(E) by the gate-voltage derivative of f(E) and integrates over energy. For a nonequilibrium distribution carrying a localized hot-carrier excess at energy E0, the gate derivative gains structure at E0, and g_m splits into a conventional density-modulation term and a shape-driven term g_m^(alpha). The shape term produces an anomalous peak whose position and drain-voltage dependence constrain the hot-carrier energy E0, spectral width sigma, and generation threshold n_c from ordinary DC or lock-in transconductance sweeps. If right, this turns standard electrical measurements into a steady-state, all-electrical spectroscopy of the out-of-equilibrium distribution, with an optional time-resolved extension that recovers the energy relaxation time.

What carries the argument

The load-bearing object is the spectroscopic identity g_m = (qW/L) ∫ j(E) ∂f(E;V_G)/∂V_G dE, where j(E) = v(E)T(E) is the spectral current kernel that preferentially weights high-energy carriers. The gate derivative of the distribution is what converts spectral structure into a measurable DC quantity: featureless for a Boltzmann tail, structured at E0 for a hot-carrier bump. Combined with the monotone-saturating generation amplitude α(V_G,V_D) = α_max [V_D²/(V_D²+V_c²)] [n/(n+n_c)], the bell-shaped dα/dn yields an analytically solvable peak position V_G^pk = V_th + (q/C_ox) n_c N_eq/(N_eq + α_max h(V_D) N_neq), and these explicit formulas are the bridge from a measured g_m(V_G) sweep to the

What would settle it

Measure g_m(V_G) at several drain voltages on a gated 2D FET, subtract the smooth density-modulation background calibrated at low V_D, and inspect the residual. The paper predicts a localized peak whose position shifts to lower V_G and whose height saturates as V_D grows; observing no peak, or a residual peak with V_D-independent position and V_D-linear height — the mobility-rolloff signature — would count against the central claim.

Watch

Extended reading notes

Core claim

The paper's central claim is that the standard reading of transconductance as a carrier-density proxy is incomplete: for a gate-independent spectral kernel j(E) = v(E)T(E), g_m equals (qW/L) times the energy integral of j(E) times the gate-voltage derivative of f(E), making it a spectral transform of the distribution shape. For a nonequilibrium distribution with a Gaussian hot-carrier excess at E0 above the transport onset, the gate derivative of f gains a bump at E0, producing a shape-driven contribution g_m^(alpha) after subtracting the smooth density background. This term peaks at V_G^pk, with the peak position encoding n_c and the spectral width sigma and the peak height sensitive to E0

Load-bearing premise

The load-bearing premise is the phenomenological saturating amplitude α = α_max [V_D²/(V_D²+V_c²)] [n/(n+n_c)] (Eq. 13): any monotone-saturating g(n) gives a peak, but the quantitative extraction of {n_c, σ, E0} from peak position and height uses this specific form, so if real hot-carrier generation is field-controlled rather than density-controlled, or fails to saturate, the predicted fingerprints would not appear.

Editorial extensions

If this is right

Ordinary transconductance sweeps can serve as steady-state spectroscopy: after calibrating the spectral kernel in equilibrium, the residual shape term's peak position and height read out E0, σ, and n_c without any optical readout.
The drain-voltage fingerprint — peak position decreasing with V_D, peak height saturating like V_D²/(V_D²+V_c²) — separates hot-carrier distribution shaping from quasi-equilibrium mobility rolloff, whose peak position is V_D-independent and whose height is V_D-linear.
An effective-temperature description is excluded by the same data: its best-fit flat g_m cannot reproduce the shape contribution, so a residual peak is direct evidence of a non-thermal, localized carrier population.
The anomaly is generic: any localized nonthermal excess above the transport onset produces the same peak, and the peak position is insensitive to the transmission and saturation functional forms tested.
A two-time-scale transient after pulsed excitation separates density relaxation from energy relaxation, making τ_E extractable from a single electrical transient when τ_n/τ_E is small enough.

Reading between the lines

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

Beyond the paper: because the peak position is governed by α(n) while the peak height is governed by the transport kernel, one could use the measured V_D-shift of the peak to test the model's central assumption that generation saturates with density, independent of the exact kernel shape.
Beyond the paper: if the spectroscopic identity holds across densities, archived g_m(V_G,V_D) data from devices never intended as spectrometers could be re-analyzed for hot-carrier signatures — a zero-cost test of the claim.
Beyond the paper: the same 'derivative of a conductance resolves distribution shape' logic could be applied to spin- or valley-polarized distributions in spin-orbit-coupled 2D systems, where the shape term would act as an all-electrical readout of distribution asymmetry.
Beyond the paper: the weakest link is the phenomenological α(n); a first-principles or Monte Carlo computation of hot-carrier generation in a specific TMD FET would either validate the saturating-density form and the quantitative extraction formulas, or show that field-controlled generation requires a modified mapping.

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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

Colleague,

This is a theory-only paper with a clean, testable idea: in a 2D FET, transconductance weights the spectral current by the gate-derivative of the distribution, so it should see the shape of a hot-carrier bump, not just its density. Working through the model, the author finds that the shape-driven piece g_m^(α) develops an anomalous peak whose position shifts and height saturates with drain voltage, and argues this fingerprint distinguishes the mechanism from ordinary mobility rolloff. The algebra is internally consistent — I checked Eqs. (16)–(20) and they reproduce the stated peak position — and the robustness checks in Appendix C show the qualitative anomaly survives changes in the transmission function and the saturation form. The discrimination from a separable mobility model is genuinely nice, and the paper is honest about its phenomenological inputs.

Where I part company is the title-level claim of 'spectroscopy.' The peak is generated by the assumed α(n,V_D) in Eq. (13), and the parameters it claims to extract — n_c, σ, E_0 — are the same ones sitting in that assumption. The peak density is n_pk = N_eq n_c/(N_eq + α_max h(V_D) N_neq), so a V_D scan constrains α_max·σ, not σ alone; the peak height couples E_0 with α_max and the kernel. The author admits this in Sec. IV.d, but the abstract and introduction still sell it as constraining {E_0, σ, n_c}. Bigger operational issue: isolating g_m^(α) from measured g_m requires subtracting the density background g_m^(n) = (W C_ox/L) vbar(α), but vbar(α) depends on the unknowns. So the background subtraction can't be done from an equilibrium calibration alone; you need the very α(n) you're trying to measure. Appendix B only fits synthetic data from the same model, which does not certify identifiability from real traces.

Read as a proposal, it's useful: the qualitative anomaly is a falsifiable prediction, and the V_D fingerprint gives a clear way to test it. Read as an established measurement technique, it's inflated. I'd send it to a referee who knows both FET transport and hot-carrier experiments, and press them on identifiability — exactly what can and cannot be extracted from a V_D sweep. If that gap closes, it could become a genuinely handy tool; as is, it's a solid mechanistic proposal with overreaching claims.

Referee Report

4 major / 4 minor

Summary. The paper proposes that the transconductance g_m = dI_D/dV_G of a 2D FET is a spectroscopic probe of the carrier distribution f(E), not merely a density meter. Within a model where the spectral current kernel j(E) is gate-independent, the authors derive g_m = g_m^{(n)} + g_m^{(α)}, with g_m^{(α)} arising from the gate-voltage dependence of a hot-carrier amplitude α(V_G,V_D). They adopt a phenomenological saturating form α = α_max [V_D^2/(V_D^2+V_c^2)] [n/(n+n_c)] and show that g_m^{(α)} develops a peak at a gate voltage V_G^pk whose position and height depend on the hot-carrier energy E_0, width σ, and generation crossover n_c. They further argue that the peak's drain-voltage fingerprint—position shift and height saturation with V_D—distinguishes it from conventional mobility-rolloff and effective-temperature models, and they add a transient extension to extract the energy relaxation time τ_E. The claims are supported by analytic expressions, numerical evaluations for MoS2-like parameters, and fits to synthetic data generated from the same model.

Significance. If the quantitative spectroscopy claim held, the paper would offer a valuable all-electrical route to hot-carrier distribution characterization in 2D FETs, with a falsifiable V_D fingerprint that separates shape effects from density and mobility effects. The algebraic framework is transparent, the decomposition g_m = g_m^{(n)} + g_m^{(α)} is clean, and the authors make an honest attempt at robustness checks against alternative saturation forms and transmission functions. However, the central quantitative claim—that peak position and height constrain E_0, σ, and n_c—is underdetermined by the paper's own equations, as detailed below. The manuscript contains no experimental data; all validation is synthetic and generated from the same model used for prediction, which cannot certify identifiability or model adequacy.

major comments (4)

[Sec. III.B and Eq. (19)] The peak position does not independently constrain σ. Equation (19) gives n_pk = N_eq n_c / (N_eq + α_max h(V_D) N_neq), with N_neq ≈ σ√π. The V_D-induced shift therefore constrains only the product α_max·σ, not σ alone, unless α_max is known. The protocol in Step 3 says fitting to Eq. (19) 'extracts n_c and σ jointly,' but α_max is introduced only later in Step 4 as a remaining free parameter, making the extraction circular. Similarly, the peak height involves E_0 only through combinations with α_max and the kernel, as the paper itself admits in Sec. IV.d. The abstract's claim that the peak position and height 'constrain E_0, σ, and n_c' is therefore not supported by the presented analysis.
[Sec. II.E, Sec. III.B, and protocol Step 1/2] Isolating g_m^(α) from a measured g_m(V_G) requires subtracting the density background g_m^(n) = (W C_ox/L) \bar{v}(α). But \bar{v}(α) depends on the same unknown α(n) via Eq. (11); the equilibrium calibration at low V_D fixes only the kernel at α=0. At high V_D the background itself is model-dependent, so the residual peak height and position depend on the assumed α(n). The synthetic-data fit in Appendix B cannot certify identifiability from real measurements because the data are generated from the same model and the background subtraction is not exercised against an independent or misspecified background.
[Sec. II.D and Appendix C] The quantitative mapping from peak position to n_c is not robust to the functional form of α(n). Equation (13) is explicitly phenomenological, and Appendix C shows that alternative saturating forms (tanh, exponential, Hill) shift the peak position slightly and alter the peak-density prefactor. Since the exact α(n) in a real device is not derived from microscopics, the extracted n_c is a model-dependent effective parameter, not a robust spectral observable. The paper should either provide a microscopic justification for Eq. (13) or reframe the extraction as yielding only parameter combinations, with a systematic uncertainty from the unknown saturation form.
[Sec. IV.b and Eq. (24)] The gate-dependent kernel correction is dismissed as a smooth background because it samples f(E) at E_b, not at E_0. However, g_m^(j) also contains \partial E_b/∂V_G, which is generally nonzero and V_D-dependent; its V_D dependence is asserted to be 'linear or only weakly in V_D' without a quantitative estimate. Since the discrimination between g_m^(α) and g_m^(j) rests on the V_D fingerprints, the paper should provide a concrete estimate or bound for g_m^(j)/g_m^(α) for the MoS2-like parameters, rather than relying on qualitative scaling.

minor comments (4)

[Abstract and Sec. I] The phrase 'locking m sweeps' in the abstract appears to be a typo for 'lock-in g_m sweeps'.
[Footnote 28] The footnote says v_T absorbs an O(1) prefactor, but the factor √2 in v(E) = √(2E/m*) is conventionally kept; this is fine, but the text should state that all reported numbers are independent of this choice, which is true because only ratios enter.
[Sec. II.D] The idealized sharp threshold creates discontinuities at V_th that are artifacts; the paper notes this, but the accompanying figure or discussion could mention how subthreshold conduction would smooth the turn-on without affecting the peak, to avoid confusion for experimental readers.
[Data availability] The data/code statement says 'available upon request.' For reproducibility of the synthetic fits and figures, depositing the code and data in a public repository would strengthen the paper.

Circularity Check

2 steps flagged · score 6.0 of 10

The anomalous-peak 'spectroscopy' is the maximum of the assumed saturating α(n) ansatz, and its validation fits synthetic data generated from the same model; the quantitative extraction of {E0, σ, nc} is therefore model-internal rather than an independent prediction.

self definitional [Sec. II.D–E, Eqs. (13), (17), (19)–(20)]
"α(VG, VD) = αmax [VD²/(VD²+Vc²)] [n(VG)/(n(VG)+nc)] ... Setting d/dn[n/(An+B)²]=0 gives the exact peak density npk = B/A = Neq nc / (Neq + αmax h(VD) Nneq), and the corresponding peak gate voltage VpkG = Vth + q npk/Cox."

The headline claim is that the peak position and height extracted from g_m sweeps constrain {E0, σ, nc}. But Eq. (19) is obtained by differentiating the assumed saturating function Eq. (13); it is the maximum of that chosen ansatz and contains exactly the same {nc, αmax, σ} parameters that were inserted as phenomenological inputs. Since Nneq ≈ σ√π, the peak shift 'encodes' the assumed Gaussian width, and the peak position reports back the assumed crossover nc. Thus the quantitative spectroscopy is an inversion of the model's own parameters, not an independent first-principles prediction. The paper's own Sec. IV.d concedes the model-assisted character and that E0 is not fixed by peak height alone.
other [Appendix B, Fig. 7 caption and text]
"Black circles are synthetic data generated from the full non-thermal model with 3 % Gaussian noise added (realistic noise level for transport measurements). ... The simultaneous fit over-constrains the model and demonstrates that spectral shape—not density or temperature alone—governs transport."

The validation offered for the quantitative parameter extraction is a fit to synthetic data generated from the same non-thermal model, with the same {E0, σ, αmax, nc} parameters, plus noise. Agreement is therefore guaranteed by construction; it certifies only internal consistency of the equations and code, not that a real measured g_m peak would determine {E0, σ, nc}. This is circular evidence for the central spectroscopic claim, particularly because the paper itself describes the extractions as 'model-assisted' and 'effective parameters' rather than model-independent quantities.

full rationale

The general identity g_m = (qW/L)∫j(E) ∂f(E;VG)/∂VG dE (Eq. 15) is simply the derivative of a spectral current integral and is not circular. The paper's self-citations, refs. [25]–[27], are contextual related work and are not load-bearing; no uniqueness theorem is imported and no ansatz is smuggled in via citation. The circularity resides in the quantitative 'spectroscopy' claim: the anomalous peak g_m^(α) is produced by differentiating the assumed saturating amplitude α(n) of Eq. (13), so Eq. (19)'s peak position is the maximum of that assumed function. Calling this peak a readout of {nc, σ, E0} is an inversion of parameters that were introduced as the phenomenological model, and the paper admits E0 enters only together with α_max and the kernel. The Appendix B fit to synthetic data generated by the same model is circular validation: it cannot establish identifiability from real devices. The qualitative existence of a peak is robust to the choice of saturating α(n) (Appendix C), so the paper has some independent qualitative content; however, the central quantitative extraction that constitutes the headline is model-internal. Score 6 reflects one or more predictions reducing by construction, without a self-citation chain.

Assumptions & free parameters 7 free parameters · 5 assumptions · 0 invented entities

The paper contributes a derivation whose output is largely organized by its own phenomenological inputs: the anomalous peak is mechanically produced by the assumed saturating α(n) (Eq. 13), with two of the three claimed 'extracted' parameters (n_c, σ) being inputs of that function. The genuinely derived content is the mapping between the assumed distribution shape and the g_m observables (position/height/V_D dependence) plus the discrimination from mobility-rolloff and T_eff models. No new entities are postulated.

free parameters (7)

E_b (transport-kernel onset energy) = 50 meV
Effective crossover energy of the spectral kernel (Eq. 5), folding contact injection, electrostatic bottlenecks, and scattering; the paper says it is 'calibrated from the equilibrium response', not derived from a microscopic barrier. All central results require E_0 > E_b.
Δ (kernel broadening) = 20 meV
Broadening of the transmission step in Eq. (5); same calibration status as E_b; sets only the magnitude, not the position, of the anomaly (App. C).
α_max (hot-carrier saturation amplitude) = 0.80
Saturation amplitude of the hot-carrier population in Eq. (13). The peak height, and thus the claimed extraction of E_0, scales with α_max; the paper admits E_0 is not fixed by the peak height alone.
n_c (hot-carrier generation crossover density) = 2.0×10^12 cm^-2
Crossover density in Eq. (13). Equation (19) shows the peak density is n_c renormalized by the model parameters, so the claimed 'extraction' of n_c from the peak position is a readback of the assumed crossover within the model. Paper concedes it is an effective, non-universal scale.
V_c (drain-voltage crossover) = 0.30 V
Sets the V_D dependence h(V_D) = V_D²/(V_D²+V_c²) in Eq. (13); the saturation fingerprint of the peak height is this assumed function; extracted by fitting peak height vs V_D.
E_0 (hot-carrier energy) = 250 meV
Center of the Gaussian hot-carrier excess (Eq. 7); the headline claim is that E_0 is 'constrained' by the peak height, but the paper concedes E_0 enters only together with α_max and the kernel, so the constraint is conditional on the model.
σ (hot-carrier Gaussian width) = 40 meV
Width of the hot-carrier bump; enters the peak-position shift through N_neq ≈ σ√π (Eq. 19). The V_D-shift of the peak is claimed to constrain σ, again within the assumed model.

assumptions (5)

standard math The drain current is an energy integral over a spectral current kernel: I_D = (qW/L)∫D(E)v(E)T(E)f(E)dE (Eq. 3).
Standard Landauer/Lundstrom-type transport formalism (refs. 12–13); the paper's starting point, unproved but conventional for mesoscale/nanoscale transport.
domain assumption Gate-independent spectral kernel: ∂j(E)/∂V_G = 0 (Sec. II.E).
Load-bearing for the spectroscopic identity Eq. (15). The paper addresses corrections in Sec. IV.b, arguing they are smooth and cannot reproduce the peak, but asserts rather than derives that ∂E_b/∂V_G ≪ 1 in the on-state.
domain assumption Nondegenerate Boltzmann statistics (Eqs. 6–7) over the modeled gate range.
At the modeled densities (n up to ~7×10^12 cm^-2, m* = 0.5 m_0, g = 4), the 2D Fermi energy reaches ~85 meV ≈ 3.3 k_BT at 300 K, so the channel is partially degenerate at the upper end of the V_G range of Fig. 3. Quantitative formulas (N_eq = k_BT, Eq. 19) would change under Fermi statistics. Not flagged in the paper.
ad hoc to paper Hot-carrier amplitude saturates with density: α = α_max h(V_D) n/(n+n_c) (Eq. 13).
Phenomenological; the paper acknowledges it is a representative form. The existence of the anomalous peak and its V_D fingerprints are algebraic consequences of monotone-saturating α(n); if a real device does not saturate on this scale, the central observable disappears.
standard math Gaussian integral N_neq ≈ σ√π for E_0 ≫ σ.
Justified for E_0 = 250 meV, σ = 40 meV (lower tail at 6.25σ).

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Pith. "Pith review of Hot-Carrier Distribution Spectroscopy by Transconductance in Two-Dimensional Field-Effect Transistors." pith.science (2026). https://pith.science/paper/4CRIKGQM

@misc{pith2026260715578,
  author       = {Pith},
  title        = {Pith review of: Hot-Carrier Distribution Spectroscopy by Transconductance in Two-Dimensional Field-Effect Transistors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4CRIKGQM}},
  note         = {Machine review of arXiv:2607.15578}
}

abstract

The transconductance $g_m = dI_D/dV_G$ of a field-effect transistor (FET) is conventionally read as a proxy for carrier density. We show that it is instead a spectroscopic probe of the carrier distribution: because $g_m$ weights the spectral current $j(E)$ by the gate-voltage derivative $\partial f(E)/\partial V_G$ and integrates over energy, it is sensitive to the \emph{shape} of $f(E)$, not merely its integrated weight $n$. We develop an energy-resolved transport framework for two-dimensional (2D) FETs and, within a gate-independent spectral-kernel approximation, derive the decomposition $g_m = g_m^{(n)} + g_m^{(\alpha)}$ into the conventional density-modulation term $g_m^{(n)}$ and a distribution-shape-driven term $g_m^{(\alpha)}$. The latter, obtained as the residual after subtracting the smooth density-modulation background from the measured $g_m$, exhibits a characteristic anomalous peak at a gate voltage $V_G^{\rm pk}$. This peak has no counterpart in equilibrium transport and \emph{cannot be explained by carrier density modulation alone}. With the spectral kernel calibrated, the peak position and height -- extracted from standard DC/lock-in $g_m$ sweeps -- constrain the hot-carrier energy $E_0$, spectral width $\sigma$, and generation threshold $n_c$, realizing a steady-state, all-electrical spectroscopy of the carrier distribution. An optional time-resolved extension further recovers the carrier relaxation time $\tau$ from the transient response following a pump excitation, establishing the 2D FET as a distribution-function spectrometer that requires no optical readout.

Figures

Figures reproduced from arXiv: 2607.15578 by the authors.

**Figure 1.** FIG. 1. Concept, device, and measurement principle (schematic). (a) A gated monolayer 2D-semiconductor field-effect transis [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** FIG. 2. (a) Carrier distribution functions [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. Main steady-state prediction, combining the current–voltage and transconductance characteristics. (a) Drain current [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: FIG. 4. Discrimination of the non-thermal distribution-shape peak from quasi-equilibrium descriptions. [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

**Figure 5.** Figure 5: FIG. 5. Two-time-scale separation in the transient drain current after a pulsed excitation (∆ [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: FIG. 6. Time-resolved response after pump excitation that injects hot carriers with [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

**Figure 7.** Figure 7: FIG. 7. Simultaneous fit of [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]

**Figure 8.** Figure 8: FIG. 8. Robustness of the [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]

Discussion (0). Continue with ORCID to comment.

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(13) uses the saturating formg(n) =n/(n+n c)

Saturation functionα(n) The phenomenological amplitude of Eq. (13) uses the saturating formg(n) =n/(n+n c). The anomalous peak in the shape-driven transconductance does not rely on this specific choice: any monotonically saturat- ingg(n) gives a bell-shapeddα/dnand hence a local- izedg (α) m peak. Figure 8(a,b) compares four represen- tative forms—the bas...
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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

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(13) uses the saturating formg(n) =n/(n+n c)

Saturation functionα(n) The phenomenological amplitude of Eq. (13) uses the saturating formg(n) =n/(n+n c). The anomalous peak in the shape-driven transconductance does not rely on this specific choice: any monotonically saturat- ingg(n) gives a bell-shapeddα/dnand hence a local- izedg (α) m peak. Figure 8(a,b) compares four represen- tative forms—the bas...

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T ransmissionT(E) The main text models the transmission by the Fermi- function barrier of Eq. (5). The anomaly does not rely on this form either. The shape-driven term en- ters through the spectral current momentsJ eq,neq =R j(E)f eq,neq(E)dEwithj(E) =v(E)T(E), and its sign and localization require only that the hot-carrier com- ponent atE 0 > Eb be trans...

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