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[Katsunori Wakabayashi](https://orcid.org/0000-0002-9147-9939)

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[Hot-carrier distribution spectroscopy by transconductance in two-dimensional field-effect transistors](https://mdr.nims.go.jp/datasets/bb219760-d63b-48ca-8630-3b58b47c4a70)

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Hot-Carrier Distribution Spectroscopy by Transconductance in Two-DimensionalField-Effect TransistorsKatsunori WakabayashiResearch Center for Materials Nanoarchitectonics (MANA),National Institute for Materials Science (NIMS), Namiki 1-1, Tsukuba 305-0044, Japan(Dated: September 10, 2026)The transconductance gm = dID/dVG of a field-effect transistor (FET) is conventionally readas a proxy for carrier density. We show that it is instead a spectroscopic probe of the carrierdistribution: because gm weights the spectral current j(E) by the gate-voltage derivative ∂f(E)/∂VGand integrates over energy, it is sensitive to the shape of f(E), not merely its integrated weight n.We develop an energy-resolved transport framework for two-dimensional (2D) FETs and, within agate-independent spectral-kernel approximation, derive the decomposition gm = g(n)m + g(α)m into theconventional density-modulation term g(n)m and a distribution-shape-driven term g(α)m . The latter,obtained as the residual after subtracting the smooth density-modulation background from themeasured gm, exhibits a characteristic anomalous peak at a gate voltage V pkG . This peak has nocounterpart in equilibrium transport and cannot be explained by carrier density modulation alone.With the spectral kernel calibrated, the peak position and height—extracted from standard DC/lock-in gm sweeps—constrain the hot-carrier energy E0, spectral width σ, and generation threshold nc,realizing a steady-state, all-electrical spectroscopy of the carrier distribution. An optional time-resolved extension further recovers the carrier relaxation time τ from the transient response followinga pump excitation, establishing the 2D FET as a distribution-function spectrometer that requiresno optical readout.I. INTRODUCTIONTwo-dimensional (2D) semiconductors such as transi-tion metal dichalcogenides (TMDs) [1–3] and their vander Waals heterostructures have emerged as a rich arenafor exploring electronic transport in reduced dimen-sions [4, 5]. Field-effect transistors (FETs) fabricatedfrom these materials simultaneously serve as testbedsfor fundamental physics and as building blocks of next-generation devices [6–10].In the standard description of FET transport, thedrain current is written asID =qWLnµVD, (1)where q is the elementary charge, W and L are the chan-nel width and length, n is the areal carrier density, µis the mobility, and VD is the source–drain voltage [11].This relation implicitly assumes that carriers obey anequilibrium or quasi-equilibrium distribution, so that µis a single scalar that summarizes all microscopic scat-tering. Under strong source–drain fields, carrier injec-tion, or optical pumping, however, the distribution func-tion acquires a hot-carrier tail whose energy is well abovethe lattice temperature. In such nonequilibrium states,the shape of the distribution—not merely its integratedweight n—governs transport, rendering Eq. (1) inade-quate [12, 13].Nonequilibrium carriers in 2D materials have beenstudied extensively by ultrafast optical spectroscopy [14].Pump–probe experiments on graphene [15] andMoS2 [16–19] reveal hot-carrier relaxation times inthe range τ ∼ 0.1–10 ps, reflecting carrier–phononscattering dynamics. Yet such optical probes arerarely combined with operando electrical gating, andthey do not directly access the transport-relevant,velocity-weighted distribution that governs ID.Here we develop an energy-resolved transport theoryfor 2D FETs that connects the shape of f(E) to measur-able device characteristics. Our starting point is a spec-troscopic reformulation of transport: within an energy-resolved transport description with a gate-independentspectral current kernel j(E), the transconductance takesthe formgm =qWL∫ ∞0j(E)∂f(E;VG)∂VGdE, (2)which makes explicit that the transconductance does notmerely reflect carrier density modulation but probes theenergy distribution of carriers. Equation (2) is itself anatural consequence of writing the current as a spectralintegral and differentiating with respect to VG. It is aclose relative of the Landauer–Büttiker formula [13, 20]and of the transport distribution function of thermo-electric transport [21]. More broadly, it instantiates ageneral principle—that a derivative of the conductanceresolves spectral structure—which underlies scanning-tunneling [22] and quantum-dot transconductance [23]spectroscopies as well as the electrical measurement ofthe quasiparticle distribution in mesoscopic wires [24].Our contribution is to apply this principle to the shapeof a nonequilibrium distribution in a 2D FET, where ityields the specific, measurable anomaly developed below.In a related line of work, the same transconductance-as-probe principle is applied to the valley degree of free-dom in WSe2, using gm to read out valley thermody-namics in multilayer films [25] and to guide valley engi-neering in bilayer gate-all-around transistors [26]. A dy-2Energy hot carrierexcess,Source DrainGateOxideField-induced hot carriergeneration(a) (b) (c)2D semiconductor channelFIG. 1. Concept, device, and measurement principle (schematic). (a) A gated monolayer 2D-semiconductor field-effect transis-tor: a gate voltage tunes the carrier density in the channel between source and drain contacts, while field-induced hot-carriergeneration drives the distribution out of equilibrium. (b) Physical mechanism (energy on the vertical axis). Left: the carrierdistribution f(E) is either thermal (dotted) or non-thermal (red), the latter carrying a localized hot-carrier excess near E0.Right: the transport kernel—an energy-selective transmission T (E) that turns on near the onset Eb and a velocity v(E) thatgrows with energy—weights high-energy carriers more heavily, so each carrier in the excess near E0 > Eb contributes morecurrent. (c) Measurement principle (schematic): a conventional, density-only response is featureless in VG (blue), and the totalnon-thermal gm (red) rises monotonically. The non-thermal signature is the shape-driven contribution g(α)m (purple, shaded),which produces an anomalous peak whose position and height read out the spectral parameters of f(E)—transconductancespectroscopy. (In the representative case shown here the total measured gm itself remains monotone; the peak is isolated ing(α)m , as quantified in Fig. 3.) The response turns on at the threshold voltage Vth [Eq. (12)], where the channel begins topopulate, and the anomaly peaks at V pkG [Eq. (20)].namical extension of the same idea uses the transconduc-tance to extract the intervalley relaxation time of WSe2transistors [27]. Corrections from a VG-dependent kernelj(E, VG) (e.g. barrier lowering, contact modulation, orband bending) are additive and discussed in Sec. IV.For a Boltzmann distribution, ∂feq/∂VG is featurelessin energy, and gm is flat—the conventional result. Fora nonequilibrium distribution with a hot-carrier peak atE0, ∂fneq/∂VG acquires structure at E0 and gm reports itas a measurable anomaly in the shape-resolved residual.The Gaussian hot-carrier peak adopted below is only aminimal model: the mechanism applies to any localizednonthermal excess δf(E) above the transport onset, notto a specific line shape. Figure 1(a) shows the device ge-ometry, (b) the spectral mechanism, and (c) the resultingmeasurement principle.Our main results are threefold. (i) The drain currenttakes the compact form ID = (qW/L)n v̄(α), with an ef-fective velocity v̄(α) set by a single parameter α thatquantifies the hot-carrier population. (ii) Within thesame spectral-kernel approximation, gm admits a trans-parent decomposition gm = g(n)m + g(α)m . The new termg(α)m produces an anomalous peak in the shape-resolvedresidual—obtained after calibrating and subtracting thesmooth density-modulation background—that cannot beexplained by carrier density modulation and directly con-strains the spectral parameters {E0, σ, nc}. This peakcarries a decisive drain-voltage fingerprint—its positionshifts to lower VG and its height saturates with VD—thatno separable, quasi-equilibrium mobility-rolloff modelcan reproduce. (iii) A time-resolved electrical measure-ment after a pump excitation yields the carrier relaxationtime τ , completing an all-electrical spectroscopy of thenonequilibrium distribution.The remainder of this paper is organized as follows.Section II develops the energy-resolved transport frame-work and derives the decomposition gm = g(n)m + g(α)m .Section III presents the anomalous g(α)m peak, its drain-voltage fingerprint, and the resulting extraction of thespectral parameters {E0, σ, nc}. Section IV places theframework in the context of prior hot-carrier work andaddresses corrections from a gate-dependent kernel, andSec. V concludes. Appendices A–C present the time-resolved extension, a fit to synthetic data, and the ro-bustness of the anomaly against modeling choices.II. MODEL AND THEORYA. Energy-resolved drain currentFor definiteness, we formulate the model for an n-type2D FET, where E ≥ 0 denotes the electron kinetic en-ergy measured from the conduction-band minimum. Thesame formalism applies to p-type devices by interpretingE as the hole kinetic energy measured from the valence-band maximum.For a 2D semiconductor of width W and length L, we3write the drain current as an integral over carrier en-ergy [12, 13]:ID =qWL∫ ∞0D(E) v(E) T (E) f(E) dE, (3)where D(E) is the density of states, v(E) is the groupvelocity, T (E) is an energy-selective transmission proba-bility, and f(E) is the carrier distribution function. Fora 2D system with parabolic dispersion, D(E) = D0 =gm∗/(πℏ2) is constant (with g the spin–valley degener-acy and m∗ the effective mass), and the group velocityisv(E) = vT√E/kBT , (4)where vT =√kBT/m∗ is a reference (thermal) velocityscale.[28] The spectral current j(E) ≡ v(E)T (E) actsas an energy filter: it weights high-energy carriers moreheavily in proportion to their velocity.We model the energy selectivity of the transmission bya smooth onset,T (E) =[1 + exp(Eb − E∆)]−1, (5)a smooth step centered at an effective onset energy Ebwith a broadening parameter ∆. We do not identify Ebwith a unique microscopic barrier height. Rather, Eb isan effective crossover energy of the transport kernel thatfolds contact injection (e.g. a Schottky barrier), electro-static bottlenecks (channel pinch-off, i.e. the saddle pointof the channel potential), and energy-dependent scatter-ing into a single energy-filtering scale. Carriers withE < Eb contribute weakly to the drain current, whilethose with E > Eb are efficiently transmitted [12]. BothEb and ∆ are thus effective transport-kernel parameters,to be calibrated from the equilibrium response v̄eq(T )rather than identified with an independent microscopicbarrier. Importantly, none of our central results dependon this specific functional form. They require only thatthe spectral current j(E) = v(E)T (E) be a smooth,monotonically increasing filter that favors high-energycarriers, so that a hot-carrier population at E0 > Eb ispreferentially transmitted. Any transmission with theseproperties yields the same anomalous g(α)m peak, in thesame spirit as the robustness against the choice of α(n)demonstrated in Appendix C.B. Carrier distribution functionsWe compare two classes of distribution, both normal-ized to the same areal carrier density n via∫∞0f(E) dE =n. Here f(E) is a carrier spectral density (carriersper unit area and energy), not a dimensionless occu-pation probability; accordingly it carries dimensions of[n]/[energy], with the density of states D0 absorbed intothe current prefactor as noted below.a. Equilibrium (Boltzmann).feq(E) = Aeq e−E/kBT , (6)where Aeq = n/Neq and Neq =∫∞0e−E/kBT dE = kBT .We adopt the Boltzmann (nondegenerate) form for ana-lytical transparency. It is the leading approximation tothe Fermi–Dirac distribution fFD(E) = {1 + exp[(E −EF )/kBT ]}−1 when the Fermi level lies below the trans-port states by more than a few kBT , i.e. in the moder-ately doped, nondegenerate regime relevant to the den-sity scale nc ∼ 1012 cm−2 considered here. For stronglydegenerate channels the Boltzmann factor e−E/kBT inEqs. (6)–(7) should be replaced by fFD(E). This changesonly the numerical values of the spectral moments Neqand Jeq, leaving the spectroscopic identity Eq. (15) andthe decomposition gm = g(n)m + g(α)m unchanged in form.The hot-carrier component, centered well above EF atE0 ≫ kBT , is nondegenerate in either description.b. Nonequilibrium (Boltzmann + hot-carrier Gaus-sian).fneq(E) = Aneq[e−E/kBT + α e−(E−E0)2/σ2], (7)where Aneq = n/(Neq + αNneq), Nneq =∫∞0e−(E−E0)2/σ2dE ≈ σ√π (for E0 ≫ σ), andα ≥ 0 controls the amplitude of the Gaussian hot-carrierbump centered at energy E0 > Eb with width σ. Theconstraint∫f(E) dE = n—where the constant D0 is ab-sorbed into the current prefactor, so that it cancels in v̄and in every gm ratio below—is satisfied by constructionvia Aneq. The Gaussian is a minimal representation of alocalized nonthermal excess at E0.Physically, such a peaked nonthermal population canbe established in several ways: high-field acceleration inthe drain-side region of a short-channel FET followedby energy-selective optical-phonon or intervalley scatter-ing; carrier injection over a contact or barrier bottleneckthat admits a narrow energy window; impact-ionization-like generation near a characteristic energy; or opticalpumping at a fixed photon energy. In each case carri-ers accumulate around a characteristic energy E0 set bythe dominant generation and relaxation bottleneck ratherthan by the lattice temperature. We do not commit toa specific microscopic mechanism; the Gaussian modelsonly the resulting localized excess. The spectroscopicmechanism developed below relies only on the presenceof such an excess well above the transport onset Eb. Thesame g(α)m anomaly therefore arises for any nonthermalexcess δf(E) that is sharply peaked in this energy range,irrespective of its detailed shape.The physical meaning of α is the relative number ofhot carriers in the Gaussian component: at α = 0 thedistribution is purely thermal, while increasing α buildsup a high-energy tail at E0 ≫ kBT .4C. Effective drift velocity and currentdecompositionSubstituting Eqs. (6)–(7) into Eq. (3) and using thelinearity of the integral, we define spectral current mo-mentsJeq =∫ ∞0j(E) e−E/kBT dE, (8)Jneq =∫ ∞0j(E) e−(E−E0)2/σ2dE. (9)The drain current then takes the compact formID =qWLn v̄(α), (10)where the effective drift velocity isv̄(α) =Jeq + αJneqNeq + αNneq. (11)Equivalently, v̄(α) =∫∞0j(E)f(E) dE/ ∫∞0f(E) dE issimply the carrier-weighted average of the spectral cur-rent j(E): the normalization Aneq and the density n can-cel between numerator and denominator, leaving eachmoment split into a thermal and an α-weighted hot-carrier part. Equation (10) preserves the form of theconventional relation ID ∝ nvdrift, but now v̄(α) isdistribution-shape-sensitive: it interpolates between thepure-Boltzmann limit v̄(0) = Jeq/Neq ≡ v̄eq and theGaussian limit v̄(∞) = Jneq/Nneq ≡ v̄neq. Because hotcarriers at E0 ≫ Eb are above the barrier (T (E0) ≈ 1)and have higher velocity (v(E0) ∝√E0), we have v̄neq ≫v̄eq, so even a small α significantly boosts ID.D. Gate-controlled carrier density andnonequilibrium parameterAbove threshold the carrier density follows the lineargate relation,n(VG) =Coxq(VG − Vth), (12)where Cox is the oxide capacitance per unit area andVth is the threshold voltage. This is an idealized sharpthreshold (n ≡ 0 below Vth, linear above), so dn/dVGjumps discontinuously at Vth. Through the decomposi-tion derived below, this produces two turn-on featuresin gm exactly at Vth: a step in the density term g(n)m ∝dn/dVG (which jumps from zero to a constant), and—inthe non-thermal case—a kink in the shape term g(α)m ∝ n(whose slope breaks where n begins to rise from zero).Both are artifacts of the idealized threshold: in a real de-vice, subthreshold conduction (n ∼ eqVG/kBT below Vth)smooths them into a gradual turn-on. In any case, thesefeatures sit at Vth and are well separated from—and donot affect—the anomalous peak at V pkG .Hot-carrierexcessFIG. 2. (a) Carrier distribution functions f(E) for the equi-librium (Boltzmann, blue dashed) and nonequilibrium (Boltz-mann + Gaussian bump, red solid, α = 0.34) cases, bothnormalized to the same carrier density n. The gray dottedcurve on the right axis shows the transmission T (E); verticaldashed lines mark the onset energy Eb and the hot-carrierpeak energy E0. (b) Spectral current integrand j(E)f(E) forboth distributions (shared energy axis with (a)). The shadedregion is the hot-carrier excess contribution to the current.Despite identical carrier densities, the nonequilibrium currentis enhanced by IneqD /IeqD ≈ 5.1 at this density, reflecting thelarger per-carrier current of the hot-carrier excess near E0(v̄neq/v̄eq ≈ 9.4 for the pure Gaussian component).The hot-carrier amplitude α depends on both VG andVD. The key qualitative property required for an anoma-lous peak in g(α)m is that α(n) rises from zero at smalln and saturates as n → ∞. A broad class of suchmonotone-and-saturating functions generically producesa localized peak in the distribution-shape contributiong(α)m derived below, because dα/dn inherits the bell shapeof the derivative of a saturating function. The qualitativepeak position, height, and VD-dependence are robust fin-gerprints of this mechanism, only weakly dependent onthe precise functional form of α(n). Alternative analyticforms (hyperbolic-tangent, exponential saturation, and aHill function in n) yield quantitatively similar g(α)m (VG)peaks, differing only in the prefactor of the peak density,as verified explicitly in Appendix C (Fig. 8).For analytical tractability we adopt the representative5phenomenological modelα(VG, VD) = αmaxV 2DV 2D + V 2cn(VG)n(VG) + nc, (13)which is proportional to the power density (through VD)and to the density of impact-ionizing or inelastically scat-tered carriers (through n). Here αmax is the saturationamplitude, Vc is the drain-voltage crossover, and nc isthe carrier-density crossover for hot-carrier generation.The high-field crossover at Vc parallels the onset of ve-locity saturation observed in monolayer MoS2 [29, 30]. Inreal devices nc is not a universal material constant butan effective crossover density set by the channel length,contact field concentration [31], dielectric environment,heat dissipation, and the dominant inelastic-scatteringpathway [32]. The value adopted below should thereforebe viewed as a representative effective scale for stronglygated TMD FETs. For n ≪ nc, α ∝ n (few carriers, fewcollisions); for n ≫ nc, α → αmaxh(VD) (saturation).We stress that Eq. (13) is a transparent phenomeno-logical interpolation, not a microscopic derivation, andthat the two factors are chosen for their limiting be-havior rather than for a specific functional form. Thedrain-voltage factor h(VD) = V 2D/(V 2D + V 2c ) representsfield-driven hot-carrier generation that switches on withthe power input at low bias and saturates once the high-field region spans the channel. Any monotonic functionrising from zero and saturating at large VD yields thesame qualitative g(α)m (VD). The density factor n/(n+nc)expresses that at low density the hot-carrier populationgrows in proportion to the number of available carriers(and hence collisions), whereas at high density it satu-rates as phase-space restriction, screening, and energy-relaxation bottlenecks limit further accumulation. Whatmatters for the anomalous peak is only that α(n) risefrom zero and saturate, so that dα/dn is bell-shaped. Weverify explicitly in Appendix C (Fig. 8) that hyperbolic-tangent, exponential, and Hill forms all produce the samepeak, with only minor quantitative differences. A first-principles kinetic treatment of α(n, VD) from the coupledhot-carrier generation–relaxation balance is beyond thescope of the present work and is left for future study.E. Transconductance as a spectral probeBefore decomposing gm, we establish its fundamentalspectroscopic content. Differentiating Eq. (3) with re-spect to VG gives, by the product rule, two contributions,gm =qWL∫ ∞0[∂j(E;VG)∂VGf(E;VG) + j(E;VG)∂f(E;VG)∂VG]dE,(14)a kernel-modulation term and a distribution-modulationterm. Within the present transport description the spec-tral current kernel j(E) = v(E)T (E) [Eqs. (4)–(5)] hasno explicit gate dependence, so ∂j/∂VG = 0, the firstterm vanishes, and we obtaingm =qWL∫ ∞0j(E)∂f(E;VG)∂VGdE. (15)The kernel-modulation term is nonzero when the gatelowers the barrier or modulates the contacts [T =T (E, VG)], induces band bending, or otherwise rendersj gate-dependent. This term is additive and is estimatedin Sec. IV. Equation (15) shows that the distribution-function contribution to gm is exactly the spectral cur-rent j(E) weighted by ∂f/∂VG: a spectrally resolved re-sponse function of the distribution. For feq, the deriva-tive ∂feq/∂VG ∝ e−E/kBT is featureless, and gm dependsonly on the total carrier density. For fneq, ∂fneq/∂VGacquires additional structure at E0 whenever α dependson VG (i.e. on n), and gm directly reports this spec-tral feature—even though no optical measurement is per-formed. This is the central spectroscopic principle of thepaper.Decomposing gm = dID/dVG explicitly via Eq. (10)with respect to VG:gm =WCoxLv̄(α)︸ ︷︷ ︸g(n)m+WCoxLndv̄dαdαdn︸ ︷︷ ︸g(α)m, (16)where we used dn/dVG = Cox/q from Eq. (12). The twoterms have distinct physical origins:• g(n)m = (WCox/L) v̄(α): the conventional density-modulation term. At equilibrium (α = 0), thisequals the constant value (WCox/L) v̄eq, giving aflat gm–VG curve in the linear regime.• g(α)m : a new distribution-shape-modulation termthat arises because the gate voltage also controlsα through n(VG). Its magnitude is proportionalto dv̄/dα = NeqNneq(v̄neq − v̄eq)/(Neq + αNneq)2,which is positive whenever v̄neq > v̄eq.The gate-voltage dependence of g(α)m is governed bydαdn=αmax h(VD)nc(n+ nc)2, (17)where h(VD) = V 2D/(V 2D + V 2c ). To find the peak ofg(α)m analytically, we substitute the α(n) dependenceinto the expression for dv̄/dα and simplify. Definingβ = αmax h(VD), one findsg(α)m ∝ n(Neq + βNneqnn+nc)2(n+ nc)2=n(An+B)2, (18)where A = Neq + βNneq and B = Neq nc. Settingd/dn[n/(An+B)2] = 0 gives the exact peak densitynpk =BA=Neq ncNeq + αmax h(VD)Nneq, (19)6and the corresponding peak gate voltageV pkG = Vth +q npkCox. (20)In the limit αmax → 0, Eq. (19) gives npk → ncand V pkG → Vth + qnc/Cox ≡ V ∗G. For finite αmax,the peak shifts to lower gate voltages by the factorNeq/(Neq + αmaxh(VD)Nneq). Since Neq = kBT andNneq ≈ σ√π, this shift encodes the hot-carrier energywidth σ. Consequently, g(α)m exhibits a peak at V pkG —ananomalous feature absent in equilibrium transport—andits position jointly reports nc (the density crossover) andσ (the hot-carrier spectral width).III. RESULTSWe evaluate Eqs. (10)–(16) numerically for parame-ters representative of a monolayer MoS2-like 2D semi-conductor (m∗ = 0.5m0, spin–valley degeneracy g = 4,T = 300 K). The gate dielectric is taken as 30 nm HfO2(εr ≈ 25), giving Cox/q = 1.4 × 1013 cm−2V−1. Thetransport-kernel onset parameters are Eb = 50 meVand ∆ = 20 meV. Hot carriers are characterized byE0 = 250 meV, σ = 40 meV, αmax = 0.80, nc =2.0×1012 cm−2, and Vc = 0.30 V. Without loss of general-ity we set Vth = 0, so that all gate voltages are measuredrelative to threshold. A finite Vth merely translates theVG axis and leaves every result unchanged.A. Carrier distributions at fixed densityFigure 2(a) compares feq(E) and fneq(E) at the samecarrier density n. The equilibrium distribution decaysexponentially with thermal energy kBT = 26 meV, whilethe nonequilibrium distribution is depressed at low en-ergies (due to normalization) but acquires a prominentGaussian bump at E0 = 250 meV.Figure 2(b) shows the spectral current integrandj(E)f(E), which determines the contribution to ID ateach energy. The equilibrium integrand is concentratedat low energies where most carriers reside, but the bar-rier T (E) strongly suppresses the contribution below Eb.The nonequilibrium integrand acquires a large additionalpeak at E0, because those hot carriers have both highervelocity v(E0) = vT√E0/kBT and unit transmissionT (E0) ≈ 1. The ratio of integrated spectral currentsis v̄neq/v̄eq ≈ 9.4, meaning each hot carrier contributes9.4× more current than a thermal carrier at T = 300 K.This value 9.4 quantifies the per-carrier current contri-bution in the pure Gaussian component (i.e., v̄neq/v̄eq),whereas the actual mixed-distribution current enhance-ment at α = α(n0, VD=0.5V), weighted by the relativepopulations of the two components, is the smaller valueIneqD /IeqD ≈ 5.1 displayed in Fig. 2(b).B. Steady-state transport predictionFigure 3(a) shows ID(VG) at VD = 0.5 V. Both curvesincrease linearly with VG, as expected from ID ∝ n, butthe nonequilibrium current is enhanced by a factor thatgrows with VG up to n ∼ nc and then saturates. This sat-uration is traced in Fig. 3(b): as n ≫ nc, α(n) saturatesat αmaxh(VD), and v̄(α) approaches a constant. The ver-tical line at V ∗G marks the crossover density n = nc. Atthis gate voltage, the current enhancement is maximalin a differential sense—the nonequilibrium ID increasesfaster with VG than the equilibrium baseline, a hallmarkof the g(α)m contribution.The central experimental prediction is shown inFig. 3(c), which plots gm(VG) together with its de-composition. The equilibrium transconductance is con-stant at (WCox/L) v̄eq above threshold, as expectedfrom conventional theory. For the representative pa-rameters used here, the total nonequilibrium gm itselfrises monotonically with VG and saturates at a higherplateau ((WCox/L) v̄(αmaxh(VD))) than the equilibriumvalue, because the density term g(n)m is monotone byconstruction and dominates the magnitude. The non-thermal signature is therefore carried not by the totalgm but by its shape-driven contribution g(α)m = gm−g(n)m ,which exhibits a pronounced anomalous peak at V pkG[Eq. (20)] and vanishes in any density-only description(where g(α)m ≡ 0). Because the density background g(n)m =(WCox/L) v̄(α) is smooth and monotone, this localizedpeak appears as a shoulder on gm(VG) and is isolated bysubtracting the calibrated monotone background. It isthis shape term—the gate-voltage derivative of the distri-bution shape—that directly encodes spectral informationbeyond n. This is the key signature of transconductancespectroscopy. With the present parameters (T = 300 K,αmax = 0.80, σ = 40 meV, VD = 0.5 V), Eq. (19) givesnpk/nc ≈ 0.38, so the peak appears at V pkG ≈ 55 mV,well below the reference scale V ∗G = qnc/Cox ≈ 143 mV.The same panel decomposes gm into its two compo-nents. The density-driven term g(n)m is a smoothly in-creasing function that inherits the VG dependence ofv̄(α), which rises monotonically as α increases. Theshape-driven term g(α)m is localized near V ∗G and vanishesboth below threshold (n → 0) and for n ≫ nc (dα/dn →0). The anomalous peak thus resides entirely in g(α)m ,while the total gm remains monotone—underscoring thatthe signature is a feature of the distribution-shape con-tribution, isolated from the smooth density background,rather than a maximum of the raw transconductance.Figure 3(d) shows the peak height of g(α)m as a functionof VD. It follows h(VD) = V 2D/(V 2D + V 2c ) (gray dashed),confirming Eq. (17). For VD ≲ Vc, g(α)m grows quadrat-ically with VD, while for VD ≫ Vc it saturates. ThisVD-tunability provides an additional experimental signa-ture: measuring gm(VG) at several VD values and fittingthe peak height to h(VD) extracts Vc, which encodes the7anomalouspeakFIG. 3. Main steady-state prediction, combining the current–voltage and transconductance characteristics. (a) Drain currentID vs gate voltage VG at VD = 0.5 V (logarithmic scale), normalized to the equilibrium saturation current. The nonequilibriumcurrent (red solid) exceeds the equilibrium baseline (blue dashed) due to the hot-carrier velocity enhancement v̄(α); the verticaldotted line marks V ∗G where n = nc. (b) Left axis: nonequilibrium amplitude α(VG), showing saturation for VG ≫ V ∗G. Rightaxis: effective velocity ratio v̄(α)/v̄eq, which reaches its maximum as α → αmaxh(VD). (c) Transconductance gm(VG) forthe equilibrium (geqm , blue solid) and nonequilibrium (total, red solid) cases, decomposed into the density-driven term g(n)m(orange dashed) and the distribution-shape-driven term g(α)m (purple dotted, filled). The equilibrium gm is flat (conventional)and the total nonequilibrium gm rises monotonically; the anomalous peak near V pkG resides in the shape contribution g(α)m(purple), isolated from the smooth monotone density background g(n)m . (d) Peak height of g(α)m vs drain voltage VD, followingh(VD) = V 2D/(V 2D + V 2c ) (gray dashed); the vertical dotted line marks Vc = 0.3 V.electric-field scale for hot-carrier generation.C. Discrimination from quasi-equilibrium modelsThe anomalous peak in the shape contribution g(α)mis the central signature of the non-thermal mechanism,but it is not, by itself, unique to it. Two quasi-equilibrium descriptions must be excluded before a peakextracted from gm(VG) can be attributed to the distribu-tion shape: a gate-dependent field-effect mobility µ(VG),and an effective-temperature (heated-Boltzmann) ensem-ble. Both fail, and the way they fail provides the decisive,falsifiable signature of the mechanism.a. Exclusion of a mobility-rolloff peak. A gate-dependent mobility µ(VG)—from surface-roughness orphonon scattering that strengthens with the verticalfield [32]—also produces a gm peak. The two are sep-arated by their drain-voltage dependence. In the linearregime any quasi-equilibrium mobility description gives8fixed/linear(separable mobility model)fails fornon-thermal excess1680FIG. 4. Discrimination of the non-thermal distribution-shape peak from quasi-equilibrium descriptions. Top row—exclusionof a mobility-rolloff peak. (a) Density map of the shape-driven transconductance g(α)m (VG, VD), normalized to the equilibriumplateau geqm ; the overlaid curve is the peak trajectory V pkG (VD), which shifts to lower VG (≈ 70 mV from VD = 0.1 to 1.0 V)while the ridge intensity saturates at large VD. (b) The two fingerprints versus VD: the peak position V pkG (red, left axis)shifts and saturates, and the peak height (purple, right axis) saturates as h(VD); a separable mobility model [Eq. (21)] insteadpredicts a VD-independent position (blue dashed) and a height linear in VD (purple dotted). Bottom row—exclusion of aneffective-temperature model (VD = 0.5 V). (c) The best-fit heated-Boltzmann Teff ≈ 1680 K (green dash-dot) reproducesID(VG) reasonably well, but only by invoking an electronic temperature far above any quasi-equilibrium heating regime. (d)The same Teff model gives a flat gm(VG), whereas the non-thermal gm rises with VG; the shaded region is the non-thermalexcess over the flat Teff prediction, unaccounted for by any effective-temperature model.ID = (qW/L)nµ(VG)VD, sogm(VG, VD) = VDqWLd[nµ(VG)]dVG≡ VD G(VG), (21)which is separable in VG and VD: the peak positionis VD-independent and the peak height scales linearlywith VD. The shape mechanism violates both. ByEqs. (19)–(20) the peak position shifts to lower gatevoltage as VD grows, while the peak height saturates ash(VD) = V 2D/(V 2D + V 2c ) [Fig. 3(d)]. For the present pa-rameters the position shifts by ∆V pkG ≈ 70 mV betweenVD = 0.1 and 1.0 V [Fig. 4(a)], and the height satu-rates above VD ∼ Vc [Fig. 4(b)]. The simultaneous po-sition shift and height saturation are not reproduced by9any separable, quasi-equilibrium mobility-rolloff model[Eq. (21)]. We note that a contrived contact or mobilityresponse with strong, nonseparable VD dependence couldin principle mimic one of the two signatures, so the dis-criminating power rests on their joint appearance ratherthan on either alone.b. Exclusion of an effective-temperature model. Pre-vious treatments of hot carriers in FETs typically intro-duce an effective electron temperature Teff > T and re-place µ by µ(Teff) [11, 12]. Such models assume thatthe heated distribution remains a displaced Maxwellian,and therefore cannot capture the Gaussian hot-carriercomponent in Eq. (7), which represents a localized non-thermal excess generated, e.g., by impact ionization, op-tical pumping at a specific photon energy, or injectionover a discrete barrier. We find the optimal Teff by min-imizing the least-squares residual in ID(VG). For thepresent MoS2-like parameters at VD = 0.5 V this givesTeff ≈ 1680 K, far beyond the regime in which a sim-ple quasi-equilibrium heating picture is physically trans-parent. Yet even at this best-fit Teff , which reproducesID(VG) reasonably well [Fig. 4(c)], the model fails for gm[Fig. 4(d)]: because gTeffm = (WCox/L) v̄(Teff) is a con-stant in VG, the model predicts a flat transconductance,whereas the non-thermal gm rises with VG and carriesthe localized shape contribution g(α)m that a heated Boltz-mann distribution cannot generate. The shaded regionin Fig. 4(d) is the resulting non-thermal excess of gm overthe flat Teff prediction—a direct fingerprint of the non-thermal character of fneq that no effective-temperaturedescription can reproduce.c. Feasibility. The anomaly is large and the discrim-inator is undemanding. The shape-driven peak reachesg(α),pkm ∼ (0.4–1.4) geqm for VD = 0.1–1.0 V—an order-unity modulation of an already routinely measured quan-tity, far above the ≲ 1% resolution of standard DC/lock-in transconductance. The diagnostic position shift ofseveral tens of mV is resolved by sub-mV gate steps,and because E0 ≫ kBT the signature survives to roomtemperature. The decisive measurement is thus a set ofgm(VG) sweeps at several VD: a peak that shifts and sat-urates identifies the shape mechanism, whereas a fixedpeak with VD-linear height identifies conventional mobil-ity rolloff.D. Two-time-scale separationBeyond the steady state, the framework predicts aspecific transient response to pulsed excitation. In thesimplest case a pump injects hot carriers at t = 0 andonly the hot-carrier amplitude relaxes, α(t) = α0 e−t/τE ,while the carrier density stays fixed. This single-time-scale pump–probe limit, which yields the energy relax-ation time τE directly from the decay of gm(t), is treatedin Appendix A. In a more general scenario—such as apulsed gate voltage or optical pump that transiently in-jects both extra carriers and hot-carrier excitations— twodistinct transients appear simultaneously:ID(t) =[n0 +∆n e−t/τn]v̄(α0 e−t/τE), (22)where τn is the carrier-density relaxation time (set by thedevice RC time or dielectric relaxation) and τE is the en-ergy relaxation time (set by carrier–phonon scattering).For τn ≪ τE , these two time scales are separated and canbe independently extracted from a single ID(t) transient.Figure 5(a) shows ID(t)/IeqD on a linear scale for τn =0.2 ps and τE = 1.0 ps. Two regimes are apparent: afast initial decay (duration ∼ τn) where the excess den-sity ∆n equilibrates, followed by a pronounced interme-diate plateau where the density has returned to n0 butα is still large. The current then decays slowly from theplateau to the equilibrium baseline over the time scaleτE . The height of the plateau encodes the amplitude α0through IplateauD = n0 v̄(α0), providing an independentdetermination of the initial hot-carrier injection level.Figure 5(b) presents the same transient on a semi-logarithmic scale, where the two exponential compo-nents appear as two linear segments with distinct slopes1/τn and 1/τE . This is the standard fingerprint of abi-exponential decay, familiar from optical pump–probestudies [15, 33], now accessible by all-electrical measure-ments. Fitting the long-time slope directly gives τE with-out contamination from the fast density dynamics, pro-vided measurements are taken at t ≳ 3τn.Figure 5(c) shows the normalized anomaly∆ID(t)/∆ID(0) for a range of separation ratiosτn/τE = 0.05–0.50. In all cases the long-time tailconverges to a common e−t/τE behavior (gray dottedreference), confirming that τE can be extracted cleanlyonce τn/τE ≲ 0.5. In optimized high-bandwidth FETgeometries the device-limited density relaxation canin principle be made substantially shorter than thecarrier–phonon energy relaxation time τE ∼ 1–5 psexpected for TMDs, placing this system within theresolvable regime. The specific bandwidth requirementsare discussed in Sec. IV.IV. DISCUSSIONa. Why transconductance spectroscopy is distinctfrom previous hot-carrier work. One might ask whetherthe present results are simply a restatement of knownhot-carrier physics in different notation. They are not,for the following reasons. First, Eq. (15) holds forany f(E), not just the Boltzmann-plus-Gaussian modelused here, provided the spectral kernel j(E) is gate-independent (corrections from a VG-dependent kernel areaddressed below). This means that gm(VG) curves mea-sured in any device, under any excitation, are spectraltransforms of the underlying distribution. Second, theanomalous peak in g(α)m cannot be reproduced by anydensity-only or quasi-equilibrium description without in-voking an ad hoc nonmonotonic VG-dependence in µ(n)10Plateau (n equil.)FIG. 5. Two-time-scale separation in the transient drain current after a pulsed excitation (∆n/n0 = 0.30, α0 = 0.60). (a)ID(t)/IeqD on a linear scale (τn = 0.2 ps, τE = 1.0 ps). The fast density decay (τn) and slow spectral decay (τE) are separatedby an intermediate plateau where n has equilibrated but hot carriers persist. (b) ∆ID(t)/∆ID(0) on a semi-logarithmic scale,revealing two linear segments with slopes 1/τn (fast, blue dashed) and 1/τE (slow, purple dashed). (c) Normalized anomalyfor τn/τE = 0.05–0.50. All curves converge to the common e−t/τE behavior (gray dotted) at long times, demonstrating cleanextraction of τE when τn/τE ≲ 0.5.or Teff(n). Indeed, a monotone µ(n) or monotone Teff(n)necessarily yields a monotone gm ∝ v̄(α(n)) (Cox/q) inthe linear regime. By contrast, the distribution-shapemechanism predicts the peak position, height, and VD-dependence simultaneously from a single unified parame-ter set {E0, σ, nc}, providing a tightly constrained, falsifi-able prediction. Third, the measurement is all-electrical:it requires only a standard lock-in gm sweep and is com-patible with any gated 2D device, with no need for pumplasers, time-resolved optics, or sub-kelvin cooling.b. Robustness against gate-dependent kernels. Thespectroscopic identity Eq. (15) was derived assuming agate-independent spectral kernel j(E) = v(E)T (E). Inreal devices, VG can also modulate Eb (barrier lowering),the contact transmission [31, 34, 35], or the band bend-ing, giving an additional kernel-modulation contributiong(j)m =qWL∫ ∞0f(E, VG)∂j(E, VG)∂VGdE, (23)so that in general gm = g(n)m + g(α)m + g(j)m .For the Fermi barrier of Eq. (5), ∂j/∂VG =−v(E) (∂T /∂E) (∂Eb/∂VG). Since ∂T /∂E is sharplypeaked at E = Eb (width ∆) with unit area, the ker-nel term reduces tog(j)m ≃ −qWL∂Eb∂VGv(Eb) f(Eb;VG). (24)Two consequences follow. First, g(j)m samples the dis-tribution at the barrier edge Eb, not at the hot-carrierenergy E0 ≫ Eb. Because the bump contributes ∝e−(Eb−E0)2/σ2 ≈ e−25 there, g(j)m carries no peak struc-ture from fneq and adds only a smooth background togm(VG)—it cannot reproduce the anomalous peak. Sec-ond, its magnitude is set by the barrier-lowering rate∂Eb/∂VG, an electrostatic/contact property that for awell-gated channel in the on-state is small (∂Eb/∂VG ≪1). The distinguishing signature of g(α)m from g(j)m istheir qualitatively different VD dependence: g(α)m scalesas h(VD) = V 2D/(V 2D + V 2c ) [Fig. 3(d)] and saturatesat large VD, while g(j)m contributions (barrier-lowering,contact modulation) at fixed VG generally scale linearlyor only weakly in VD. Measuring the VD-dependenceof the anomalous peak thus provides an in-situ separa-tion of distribution-shape and kernel-modulation mecha-nisms. In practice, since contact and barrier-modulationeffects can themselves exhibit nonlinear VD behavior, themost robust criterion is not a single VD-dependence alonebut the simultaneous consistency of peak position, peakheight, and the joint fit to ID and gm (Appendix B,Fig. 7)—the same over-constrained signature that ex-cludes the Teff description.c. Other device-related mechanisms and the over-constrained criterion. We emphasize that the appear-ance of a single peak in gm(VG) is not, by itself, claimedas a unique fingerprint of hot carriers. Contact resistanceand its bias dependence, interface traps, self-heating,field- or density-dependent mobility degradation, andgate-induced barrier modulation can each produce non-monotonic features in the raw transconductance. Whatdiscriminates the distribution-shape mechanism is notthe peak alone but the joint, over-constrained behaviorof several observables that these parasitic effects do notreproduce together: (i) the peak position V pkG shifts tolower gate voltage with increasing VD [Eqs. (19)–(20)],whereas a separable mobility or trap response gives aVD-independent position; (ii) the peak height saturatesas h(VD) rather than growing linearly in VD; (iii) a singlespectral parameter set {E0, σ, αmax, nc} must reproduceID(VG) and gm(VG) simultaneously (Appendix B); and(iv) the low-bias limit VD ≪ Vc must recover the flat, cal-11ibrated equilibrium transconductance with V pkG → V ∗G.Contact, trap, and self-heating artifacts are, moreover,suppressible or separable by standard controls—four-probe or transfer-length contact de-embedding, gate-bias- and frequency-dependent trap characterization, andpulsed (low-duty-cycle) measurements that eliminatesteady-state self-heating. The mechanism should there-fore be regarded as identified by this diagnostic protocolas a whole, not by any single peak.d. Role of inelastic scattering and the transmis-sion kernel. The energy-selective transmission T (E) inEq. (3) is an effective, elastic-like energy filter calibratedfrom the equilibrium response; it does not resolve indi-vidual inelastic collisions. Inelastic scattering enters ourdescription not through T (E) but through the distribu-tion itself—its steady-state hot-carrier amplitude α and,in the transient case, its relaxation law α(t) = α0e−t/τE .The steady-state spectroscopy is therefore independentof the microscopic collision integral: it reads out what-ever nonthermal f(E) is present, however generated. Thetime-domain extension (Sec. III; Appendix A) treats en-ergy relaxation as a slow evolution of this distribution pa-rameter relative to the instantaneous transport response,which is justified when the transit/RC time is short com-pared with τE . It is not a microscopic transport theoryof energy-nonconserving events during a single traversal,and it would break down if inelastic mean free paths be-came comparable to the channel length, so that T itselfacquired a strong, energy-redistributing bias dependence.That regime lies beyond the present kernel approxima-tion.e. Over-constrained simultaneous fit. Beyond ex-cluding the effective-temperature description (Sec. III C),the non-thermal model provides an over-constraineddemonstration: a single parameter set {E0, σ, αmax, nc}simultaneously reproduces both gm(VG) and ID(VG),whereas the Teff model adds one free parameter yet al-ready fails at the first observable (gm). A quantitativesimultaneous fit to synthetic data, confirming that thespectral shape of f(E)—not its integrated density or ef-fective temperature—governs transport, is given in Ap-pendix B (Fig. 7).f. Spectroscopic information from the gm anomaly.Equations (19)–(20) show that the peak position V pkGconstrains two quantities: (i) nc, extracted from V pkG inthe weak hot-carrier limit (VD ≪ Vc) where npk → nc;and (ii) the Gaussian width σ (via Nneq ≈ σ√π), ex-tracted from the shift ∆V pkG (VD) = V ∗G − V pkG (VD) as afunction of VD. Independently, the peak height g(α),pkm ∝JneqNeq−JeqNneq ∝ v̄neq− v̄eq is sensitive to E0 throughv(E0) = vT√E0/kBT , providing the hot-carrier peak en-ergy. By performing gm(VG) measurements at multipleVD values one can further extract Vc, and temperature-dependent sweeps give Eb and ∆ from v̄eq(T ). Takentogether, a single set of gm–VG–VD measurements atvaried T constrains the parameters {nc, σ, E0, Vc, Eb,∆},complementary to optical techniques that typically ac-cess only one or two of these simultaneously. These ex-tractions are model-assisted: they presuppose the cali-brated spectral-kernel form (Step 1) and a localized ex-cess, so the inferred values are best regarded as effec-tive parameters of that description rather than uniquelymodel-independent quantities. In particular, E0 entersthe peak height only together with αmax and the kernel,and is not fixed by the height alone.g. Two-time-scale separation as the third experimen-tal pillar. Equation (22) establishes a hierarchy ofelectrical measurements for complete spectroscopy off(E). The steady-state gm(VG) sweep (Pillar I: Fig. 3)constrains the spectral parameters {E0, σ, nc} throughthe peak position and height. The over-constrainedconsistency between gm and ID (Pillar II) excludesany effective-temperature description in the main text[Fig. 4(c,d)]. A quantitative simultaneous fit to syn-thetic data, which verifies that the non-thermal modelis strongly constrained by the joint gm–ID data, is pro-vided as a supplementary demonstration in Appendix B(Fig. 7). The two-time-scale transient (Pillar III: Fig. 5)then isolates the energy relaxation time τE from the den-sity relaxation time τn in a single ID(t) measurement byexploiting the bi-exponential structure of Eq. (22). To-gether these three observables constrain the full param-eter set {E0, σ, nc, τE} within the calibrated model fromelectrical measurements alone, without relying on opticalspectroscopy for readout. Whereas Pillars I and II relyon standard DC/lock-in sweeps, Pillar III requires GHz-to-THz-bandwidth time-domain instrumentation and istherefore best viewed as a specialized extension that com-plements the steady-state observables rather than replac-ing them.h. Experimental measurement protocol. We outlinea self-contained protocol by which the spectroscopic pa-rameters can be extracted sequentially from standardelectrical measurements. The minimum experimentaldataset for verifying the non-thermal mechanism and ex-tracting {nc, σ, Vc, E0, αmax} is a set of gm(VG) traces atseveral VD values (Steps 2–4 below), which requires onlya standard DC/lock-in FET sweep. The time-domainextension (Step 5) is optional and yields the additionalparameter τE .Step 1 — equilibrium calibration. At low drain voltageVD ≪ Vc, measure ID(VG) and gm(VG) at several tem-peratures T = 200–350 K. The transconductance is flatand equals (WCox/L) v̄eq(T ), giving the equilibrium ve-locity v̄eq(T ), which follows the thermally averaged spec-tral kernel and increases with temperature. Fitting thistemperature dependence to Eqs. (8)–(11) with α = 0 ex-tracts the transport-kernel onset parameters Eb and ∆.Step 2 — hot-carrier signature. Increase VD to val-ues comparable to or exceeding Vc. Subtract the smoothequilibrium-calibrated density background g(n)m from themeasured gm(VG). If the residual shape contribution g(α)mdevelops a localized peak, the effective-temperature de-scription has broken down and the non-thermal modelis required. The absence of such a peak at the same12VD and T would instead confirm quasi-equilibrium con-ditions, providing a built-in null test. Repeat at severalVD values (VD = 0.2–0.7 V recommended) to track thepeak height as a function of VD.Step 3 — VD-dependent peak height. Plot the peakheight g(α),pkm (VD) against VD. A fit to h(VD) =V 2D/(V 2D + V 2c ) [Eq. (17)] extracts Vc, the electric-fieldcrossover for hot-carrier generation. The peak positionV pkG (VD) simultaneously yields npk(VD) via Eq. (12); fit-ting to Eq. (19) with Vc known then extracts nc and thespectral width σ jointly.Step 4 — simultaneous fit and falsification. Using theparameters {Eb,∆, Vc, nc, σ} from Steps 1–3, computeID(VG) and gm(VG) at all measured VD values. The non-thermal model should reproduce both observables withstrongly reduced parameter freedom: only E0 and αmaxremain as global fit parameters, both already constrainedby Step 2. As a falsification test, confirm that the best-fit Teff model gives a flat gm(VG) and fails to capture theshape contribution g(α)m [Fig. 4(d)].Step 5 — time-domain extraction of τE. Apply a fastelectrical pulse (gate or drain) with a sub-nanosecondto picosecond-scale rise time and record ID(t) withhigh-bandwidth (GHz-to-THz) instrumentation. Fit thebi-exponential transient [Eq. (22)] to extract τn (thefast, density-relaxation component, set by device RC)and τE (the slow, spectral-relaxation component, setby carrier–phonon scattering). The full parameter set{E0, σ, nc, Vc, Eb,∆, τE} is then constrained within thecalibrated model from all-electrical measurements.i. Applicability and experimental scales. Themethod applies whenever a localized nonthermal excessis separated from the transport onset by more than thethermal broadening, E0 − Eb ≳ few × kBT , so that thehot-carrier feature is not washed out by the Boltzmanntail. For the representative TMD parameters used here(Eb = 50 meV, E0 = 250 meV) this is comfortablysatisfied at room temperature (kBT ≈ 26 meV), andthe signature therefore survives to 300 K and above.Lowering the temperature sharpens the spectral res-olution but can enhance competing trap and contacteffects; raising it broadens both the thermal backgroundand the hot-carrier excess. In gate voltage, the relevantdensity scale is n ∼ nc ∼ 1012 cm−2, reached routinely instrongly gated, high-κ TMD FETs, and short channels(L ∼ 10–100 nm) maximize the high-field region thatgenerates the excess. The steady-state observables(Pillars I and II) require only standard DC/lock-ininstrumentation, whereas the optional time-domainextraction of τE (Pillar III) assumes energy-relaxationtimes τE ∼ 0.1–10 ps typical of TMDs and henceGHz-to-THz measurement bandwidth. Outside theseranges—strongly degenerate channels, E0 comparable toEb, or dominant elastic disorder or spatial inhomogeneity(e.g. threshold variation or contact nonuniformity) thatsmears the onset—the shape contribution persists inprinciple but its peak may be too weak or too broad toresolve.j. Material and geometry considerations. Thesignal-to-noise ratio for detecting g(α)m is maximizedwhen v̄neq/v̄eq is large, which requires E0 ≫ Eb andE0 ≫ kBT . TMDs with high-κ gate dielectrics (largeCox) shift V ∗G to more accessible gate voltages. Short-channel devices (L ∼ 10–50 nm) increase the high-fieldregion and hence α at a given VD. For the pump–probetime-domain experiment, the bandwidth requirement is∼ 1/τ ∼ 1 THz for TMD materials [36], suggesting thatTHz near-field probing or GHz on-chip circuitry couldrealize this measurement.V. CONCLUSIONWe have established that the transconductance gm ofa 2D FET is fundamentally a spectroscopic quantity. Itis the spectral current j(E) weighted by the gate-voltagederivative of the carrier distribution f(E) and integratedover energy. As such, it encodes the shape of f(E), notmerely its integrated density. This spectroscopic identity[Eq. (15)] is exact for a gate-independent spectral kernel.It gives rise to an anomalous peak in the distribution-shape contribution g(α)m , isolated from the smooth mono-tone density background of the measured gm. This peak(i) has no counterpart in equilibrium transport, (ii) can-not be reproduced by any quasi-equilibrium descriptionwithout invoking ad hoc nonmonotonic VG-dependencein transport coefficients, and (iii) constrains the spectralparameters of the hot-carrier distribution through its po-sition, height, and drain-voltage dependence.Two steady-state observables, accessible with standardDC/lock-in instrumentation, provide the primary exper-imental verification of the non-thermal character of thedistribution. Pillar I is the anomalous peak in the shapecontribution g(α)m (Fig. 3). Its decisive drain-voltagefingerprint—a peak that shifts to lower VG and saturatesin height with VD—distinguishes it from a conventionalmobility-rolloff peak, which would instead remain fixed inposition and grow linearly with VD [Fig. 4(a,b)]. Pillar IIis the exclusion of any effective-temperature description,which gives a flat gm(VG) and fails to reproduce theshape contribution g(α)m even at its best fit [Fig. 4(c,d)].The over-constrained simultaneous fit of both ID andgm with a single parameter set {E0, σ, αmax, nc} pro-vides a supplementary quantitative confirmation (Ap-pendix B, Fig. 7). Together these constrain the ef-fective spectral parameters {E0, σ, nc, Vc, Eb,∆} withoutoptical spectroscopic readout. The energy relaxationtime τE becomes accessible through a more demand-ing time-domain extension—the bi-exponential transientID(t) (Pillar III, Fig. 5)—which separates the energy re-laxation from the density relaxation τn but requires GHz-to-THz-bandwidth instrumentation.The transconductance provides a spectroscopic probe ofcarrier distributions, revealing non-thermal features thatlie beyond the reach of any quasi-equilibrium description13without ad hoc nonmonotonic gate-voltage dependence.This capability positions gated 2D semiconductors as on-chip distribution-function spectrometers, opening path-ways to (i) systematic mapping of hot-carrier dynamicsacross phonon channels, valley degeneracies, and tem-peratures in van der Waals monolayers; (ii) engineer-ing of g(α)m peaks via dielectric and channel geometry forenergy-selective carrier amplification; and (iii) extendingtransconductance spectroscopy to multi-band and spin-orbit-coupled 2D systems where distribution-shape ef-fects couple to valley and spin degrees of freedom.ACKNOWLEDGMENTSThe author thanks S. Adhikary and T. Kameda foruseful discussions. This work was supported by JSPSKAKENHI (Grants No. JP25K01609, No. JP22H05473,and No. JP21H01019).DATA AVAILABILITYThe data that support the findings of this study, andthe code used to generate the figures, are available fromthe corresponding author upon reasonable request.Appendix A: Pump–probe transient at fixed carrierdensityThe two-time-scale analysis of the main text reduces,in the limit of a fixed carrier density, to a single-exponential pump–probe transient that isolates the en-ergy relaxation time τE . Consider a pump (optical orelectrical) that injects hot carriers at t = 0, establishingα(t=0) = α0. For t > 0 the hot carriers relax via carrier–phonon scattering, α(t) = α0 e−t/τE , with τE the energyrelaxation time. In a pump–probe geometry, the gate isheld fixed at VG = V ∗G and a small AC modulation δVGprobes the instantaneous transconductance. Because thepump-generated α is externally imposed and does notdepend on VG, only the density-driven term survives:gm(t) =WCoxLv̄(α0e−t/τE). (A1)Figures 6(a) and (b) show the resulting transients inID(t) and gm(t) for τE = 1 ps, normalized to their equi-librium values. Both decay monotonically from theirpump-excited values back to equilibrium. The decay isexponential in α(t) but not strictly exponential in ID orgm, because v̄(α) is nonlinear in α. Fitting the tran-sient to Eq. (A1) therefore provides τE and simultane-ously v̄neq/v̄eq, a ratio that encodes E0 (the hot-carrierenergy) via Eqs. (8)–(11).Figure 6(c) shows normalized transients∆gm(t)/∆gm(0) for several relaxation times τE .The curves are distinguishable over a decade in τEwithin a 6 ps window, demonstrating that the mea-surement discriminates phonon-scattering channelswith sub-picosecond resolution, consistent with time-resolved Raman measurements of longitudinal-optical(LO) phonon populations in III–V semiconductors [33].Compared to optical pump–probe techniques [15, 16]and recent electrical photoresponse measurements invan der Waals heterostructures [37], the present ap-proach directly accesses the transport-weighted carrierdistribution rather than the optical spectral weight orthe transient photocurrent, and can be combined withelectrostatic gating to tune n and V ∗G in situ.Appendix B: Simultaneous fit to synthetic dataFigure 7 demonstrates quantitatively that the non-thermal model fneq [Eq. (7)] with the single parame-ter set {E0, σ, αmax, nc} simultaneously reproduces bothgm(VG) [panel (a)] and ID(VG) [panel (b)]. The twoobservables probe fundamentally different moments off(E): gm is sensitive to ∂f/∂VG and therefore to theshape of f , while ID tracks the total spectral weight.A model that fits both with a single parameter set istherefore over-constrained, providing unambiguous evi-dence that the spectral shape of f(E) governs transport,not merely its integrated density or effective tempera-ture. The Teff model cannot achieve this: it adds onlyone free parameter yet already fails at the first observable(gm).Appendix C: Robustness against modeling choicesThe two phenomenological ingredients of the model—the saturation function g(n) in the hot-carrier amplitudeα(n) and the energy-selective transmission T (E)—arenot derived from first principles. We show here that theanomalous g(α)m peak is insensitive to both choices.1. Saturation function α(n)The phenomenological amplitude of Eq. (13) uses thesaturating form g(n) = n/(n + nc). The anomalouspeak in the shape-driven transconductance does notrely on this specific choice: any monotonically saturat-ing g(n) gives a bell-shaped dα/dn and hence a local-ized g(α)m peak. Figure 8(a,b) compares four represen-tative forms—the baseline hyperbola n/(n + nc), a hy-perbolic tangent tanh(n/nc), an exponential saturation1−e−n/nc , and a Hill function n2/(n2+n2c)—all normal-ized to rise from zero and saturate on the scale n ∼ nc[Fig. 8(a)]. The resulting g(α)m (VG) curves [Fig. 8(b)],each normalized to its own peak, all exhibit a localizedpeak near V ∗G with comparable width. The forms with140 1 2 3 4 5Time t/ E123456I D(t)/Ieq D(a)ID(t) at VG = V *GIeqD0 1 2 3 4 5Time t/ E123456g m(t)/geq m(b)gm(t) (pump probe)geqm0 1 2 3 4 5 6Time t (ps)0.00.20.40.60.81.0g m(t)/g m(0)(c)E = 0.5 psE = 1.0 psE = 2.0 psE = 5.0 pse t/ E (linear)FIG. 6. Time-resolved response after pump excitation that injects hot carriers with α0 = 0.6 at t = 0, for τE = 1 ps. (a) Draincurrent transient ID(t)/IeqD at VG = V ∗G. (b) Transconductance transient gm(t)/geqm , which decays from the nonequilibriumvalue back to equilibrium as hot carriers relax. (c) Normalized transconductance anomaly ∆gm(t)/∆gm(0) for relaxation timesτE = 0.5, 1.0, 2.0, and 5.0 ps (blue, green, red, purple). The gray dotted line shows the pure exponential e−t/τE for comparison,illustrating the slight nonlinearity due to v̄(α).0.0 0.1 0.2 0.3 0.4 0.5Gate voltage VG (V)0.00.20.40.60.81.01.2g m (normalized)Peak: g( )m(a)DataTeff fitfneq fit (total)g( )m  (shape)0.0 0.1 0.2 0.3 0.4 0.5Gate voltage VG (V)0.00.20.40.60.81.01.2I D (normalized)Same {E0, , max, nc} as (a)(b)DataTeff fitfneq fitFIG. 7. Simultaneous fit of gm(VG) and ID(VG) using the non-thermal model fneq (red dashed) versus the best-fit Teff ≈ 1680 Kmodel (green dash-dot). Black circles are synthetic data generated from the full non-thermal model with 3% Gaussian noiseadded (realistic noise level for transport measurements). (a) gm(VG): the non-thermal fit reproduces the data, including theshape contribution g(α)m (purple dotted) that carries the anomalous peak; the Teff model gives a flat response with no suchshape term. (b) ID(VG): both models approximate the current, but only fneq uses the same four parameters as panel (a). Thesimultaneous fit over-constrains the model and demonstrates that spectral shape—not density or temperature alone—governstransport.a steeper onset (tanh, Hill) place the peak slightly lowerin VG and decay faster at large VG, but the qualitativesignature—a localized g(α)m peak driven purely by the dis-tribution shape—is common to all of them. The peak po-sition is therefore a robust fingerprint of the saturatinghot-carrier generation mechanism rather than an artifactof the particular analytic model adopted in the main text.2. Transmission T (E)The main text models the transmission by the Fermi-function barrier of Eq. (5). The anomaly does notrely on this form either. The shape-driven term en-ters through the spectral current moments Jeq,neq =∫j(E)feq,neq(E) dE with j(E) = v(E)T (E), and its signand localization require only that the hot-carrier com-ponent at E0 > Eb be transmitted more efficiently per15FIG. 8. Robustness of the g(α)m peak against the two phenomenological modeling choices. Top row—saturation function α(n).(a) Four saturating forms of g(n) versus gate voltage: hyperbolic n/(n+nc) (blue), tanh(n/nc) (red dashed), 1−e−n/nc (greendash-dot), and the Hill function n2/(n2 + n2c) (purple dotted). (b) Corresponding g(α)m (VG), each normalized to its own peak:the peak position stays near V ∗G for every form. Bottom row—transmission T (E). (c) Four transmission forms rising from 0 to1 across Eb: Fermi step (blue), error-function (red dashed), linear ramp (green), and the discontinuous hard step Θ(E − Eb)(purple dotted); E0 marks the hot-carrier energy. (d) Corresponding g(α)m (VG), normalized to the equilibrium plateau geqm : thepeak occurs at the same gate voltage for all forms (position set by α(n), not T ); only the height varies.carrier than the thermal component, i.e. v̄neq > v̄eq.Figure 8(c,d) compares four transmission forms that allrise from 0 to 1 across Eb—the baseline Fermi step[1 + e(Eb−E)/∆]−1, an error-function step, a linear ramp,and the discontinuous hard step Θ(E − Eb) [Fig. 8(c)].The resulting g(α)m (VG), normalized to the equilibriumplateau geqm of the same form [Fig. 8(d)], peaks at thesame gate voltage for every transmission—the peak po-sition is fixed by α(n), not by T (E)—while its heightvaries only mildly, remaining of order the equilibrium re-sponse (1.3–1.8 geqm ) even for the discontinuous hard step.The transmission shape thus sets the magnitude of theanomaly but neither its existence nor its location. Takentogether, the two robustness checks are complementary:the peak position is set by α(n) and is insensitive toT (E), while the peak height is set by T (E) and the hot-carrier energy—so neither phenomenological input con-trols the existence of the signature.16Appendix D: Extension to linear (Dirac) dispersionThe framework does not rely on the constant densityof states of a parabolic 2D band. For a linear (Dirac)dispersion E = ℏvF |k|, the density of states is D(E) =g E/(2πℏ2v2F ) ∝ |E| and the band velocity is energy-independent, v(E) = vF . Writing the drain current inthe same energy-resolved form [Eq. (3)], the transportnow enters through the spectral kerneljD(E) = D(E) vF T (E) ∝ E T (E), (D1)which replaces the parabolic-band kernel j(E) =v(E)T (E) ∝√E T (E). In both cases j(E) is a smooth,monotonically increasing energy filter that favors high-energy carriers, so a localized nonthermal excess atE0 > Eb is still weighted more heavily than the ther-mal population. 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