# Fileset [main.pdf](https://mdr.nims.go.jp/filesets/dad82eae-3455-47ee-a1dc-107ddbffd4fe/download) ## Creator [Katsunori Wakabayashi](https://orcid.org/0000-0002-9147-9939) ## Rights ©2026 American Physical Society[In Copyright](http://rightsstatements.org/vocab/InC/1.0/) ## Other metadata [Electrical spectroscopy of intervalley relaxation in

{WSe}_{2}

transistors](https://mdr.nims.go.jp/datasets/412d9a2d-b7bc-4baf-aeb3-bf187d24b60b) ## Fulltext Electrical Spectroscopy of Intervalley Relaxation in WSe2 TransistorsKatsunori Wakabayashi1, 2, ∗1Research Center for Materials Nanoarchitectonics (MANA),National Institute for Materials Science (NIMS), Tsukuba 305-0044, Japan2Kwansei Gakuin University, Gakuen-Uegahara 1, Sanda 669-1330, Japan(Dated: August 18, 2026)We show that the transconductance of multilayer WSe2 field-effect transistors serves as a di-rect electrical spectrometer of the intervalley relaxation time τiv, previously accessed mainly byultrafast optical techniques. Extending an equilibrium valley-thermodynamics framework with asingle relaxation equation for the Γ-valley carrier fraction fΓ(t), we predict three signatures: (i) aLorentzian transconductance gm(ω) = gm,0 + g0m,v/(1 + iωτiv), whose imaginary part peaks atωc = τ−1iv with opposite signs for bilayer and trilayer; (ii) a two-stage current transient after a gatestep, exhibiting bilayer overshoot or trilayer undershoot; and (iii) sweep-rate-proportional hysteresiswhose gate-voltage profile and the layer-dependent transconductance sign reversal distinguish valleyfrom trap-induced dynamics. All three signatures provide quantitative electrical access to τiv withstandard radio-frequency (rf) and direct-current (dc) instrumentation.I. INTRODUCTIONA transistor channel hosting multiple valleys with dif-ferent effective masses carries a slow internal degree offreedom: the intervalley carrier distribution. When theintervalley equilibration time τiv is not negligibly shortcompared with the gate-modulation period, this distribu-tion cannot follow the gate adiabatically, imprinting itselfon all time-dependent transport observables. The conse-quences form a recognizable set of dynamic signatures —frequency-dependent transconductance, two-stage cur-rent transients, and sweep-rate-proportional hysteresis— whose gate-voltage profiles and the layer-dependenttransconductance sign reversal distinguish valley dynam-ics from charge-trapping effects. Accessing τiv throughstandard rf and dc instrumentation would provide a di-rect spectroscopic route to an internal scattering chan-nel previously accessed mainly through ultrafast opticalspectroscopy.Multilayer WSe2 provides a layer-tunable platform forthe study of these effects [1–8]. The valence-band max-imum shifts from the K point in the monolayer to-ward the Γ point with increasing layer number, drivenby interlayer coupling and spin-orbit interaction [9–13],placing ∆KΓ ≈ kBT near room temperature in thebilayer. This competition enables gate-tunable redis-tribution between the K and Γ valleys, producing avalley-dependent transconductance contribution gm,v =−∆µ (W/L)VDSqpχv parameterized by the valley sus-ceptibility χv ≡ ∂fΓ/∂VGS , completing the decomposi-tion gm = gm,0+ gm,v [14–19]. Such treatments assumedquasi-static valley equilibrium. The relaxation time τivin WSe2 is set by phonon-mediated scattering at roomtemperature (sub-ps to a few ps) [20–26] but can be sub-stantially longer at reduced temperatures or under gate-induced non-equilibrium conditions [27–29], potentially∗ Email Address: WAKABAYASHI.Katsunori@nims.go.jpplacing ωc = τ−1iv in the accessible MHz–GHz window.To date, τiv in WSe2 has been probed almost exclusivelyby ultrafast optical spectroscopy [30, 31], which requiresspecialized laser and optical cryostat infrastructure be-yond conventional electrical transport setups.In this paper, we extend the equilibrium valley frame-work to the non-equilibrium regime through a single re-laxation equation for fΓ(t), characterized by the inter-valley relaxation time τiv. This minimal extension pre-dicts three measurable consequences of delayed interval-ley redistribution: (i) a Lorentzian frequency dependenceof gm(ω) with characteristic frequency ωc = τ−1iv ; (ii) atwo-stage current transient after a gate step exhibitingbilayer overshoot or trilayer undershoot, determined bythe sign of g0m,v; and (iii) sweep-rate-proportional hys-teresis whose gate-voltage profile follows qpχv(VGS) andreverses sign between the two layer numbers. In eachcase, the transconductance sign reversal with layer num-ber and the gate-voltage dependence allow valley-origindynamics to be distinguished from charge-trapping ef-fects, and together they provide a quantitative electricalroute to τiv. Figure 1(a) shows the device geometry andmeasurement configuration, Fig. 1(b) illustrates the re-sulting valley-population lag, and Fig. 1(c) displays thepredicted spectroscopic signature in Im[gm(ω)].II. DYNAMIC VALLEY MODELThe equilibrium valley-thermodynamics frame-work [15] yields a transconductance decompositiong(0)m = gm,0 + g0m,v, where gm,0 is the conventionalcharge-accumulation term andg0m,v = −WLVDS ∆µ qpχv, (1)with ∆µ = µK − µΓ > 0, q the elementary charge, p thetotal hole density, and χv ≡ ∂fΓ/∂VGS the valley suscep-tibility. The sign of g0m,v follows that of ∆KΓ: negativefor the bilayer, positive for the trilayer.mailto:WAKABAYASHI.Katsunori@nims.go.jp2Source DrainGate(top)Si(back gate)hBNLock-inamplifier(a) Device and measurement scheme (b) Valley population dynamics (c) Transconductance spectroscopyGate modulationValley population in Γ valley01measurement window(phase delay)(actual)(equilibriuminstantaneous) 4 105 106 107 108-101103 10 109 1010BilayerTrilayerFrequencyValley contribution to transconductanceAC gate modulation and transconductance detectionFIG. 1. Overview of transconductance spectroscopy of intervalley relaxation. (a) Schematic of a multilayer WSe2 transistor indual-gate geometry with hexagonal boron nitride (hBN) dielectric [32, 33]. An alternating-current (ac) gate voltage VGS(t) =V0 + δV eiωt drives the device; the drain current ID(ω) is detected by a lock-in amplifier to extract gm(ω) = dID(ω)/dVGS(ω).(b) Valley population fΓ(t) in the Γ valley (solid) compared to the instantaneous equilibrium value (dashed), illustratingthe phase delay ∆t = ϕ/ω arising from the finite intervalley relaxation time τiv. (c) Predicted imaginary part of the valleytransconductance Im[gm(ω)] for bilayer (blue) and trilayer (red) WSe2. Opposite-sign Lorentzian peaks centered at ωc = τ−1ivprovide a direct spectroscopic readout of τiv.undershootovershootvalley relaxationFIG. 2. Two-stage current response to a gate step ∆VGS =0.2V (upper panel) for bilayer (blue) and trilayer (red) WSe2(lower panel, thick lines). Thin dashed lines (fΓ = fi, frozen)show the valley-frozen reference: charge responds instanta-neously (τRC ≪ τiv) while fΓ remains at its pre-step value fi.Valley relaxation then drives the current toward ID(∞): over-shoot for the bilayer and undershoot for the trilayer, with acharacteristic timescale τiv. The opposite-sign departures re-flect sgn(g0m,v) reversing between layers.The equilibrium fraction of holes occupying the Γ val-ley at gate voltage VGS isf eqΓ (VGS) =peqΓ (VGS)peqK (VGS) + peqΓ (VGS), (2)where peqν is the equilibrium two-dimensional hole den-sity in valley ν ∈ {K,Γ}, determined from Fermi–Diracstatistics with valley density of states Dν ∝ m∗ν and thegate-controlled Fermi energy [15]. The susceptibility χvpeaks near theK–Γ band-crossing condition and vanishesin both the subthreshold (p → 0) and heavily accumu-lated limits, confining the valley contribution g0m,v ̸= 0to the above-threshold partially occupied regime.Layer-dependent sign reversal.— The sign of χv—andhence of g0m,v—is determined by which valley constitutesthe valence-band maximum, a property that reverses signwith layer number in WSe2 [9, 11]. In the bilayer (∆KΓ =+26meV), the K valley lies at higher hole energy: holesat low carrier density reside predominantly in K, and apositive gate step drives them progressively into Γ, givingχv > 0 and hence g0m,v < 0. In the trilayer (∆KΓ =−49meV), the ordering reverses: holes first fill Γ, andthe Γ fraction decreases as K also becomes occupied withincreasing gate voltage, giving χv < 0 and g0m,v > 0. Thislayer-dependent transconductance sign reversal is absentin single-valley materials or in systems where ∆KΓ hasa fixed sign, and is specific to the layer-dependent bandstructure of multilayer WSe2.We extend this to the non-equilibrium regime witha single relaxation equation for the Γ-valley fractionfΓ(t) = pΓ(t)/p(t):dfΓdt=f eqΓ [VGS(t)]− fΓ(t)τiv, (3)3the transistor analogue of Debye relaxation in di-electrics [34]. Extensions to distributed τiv (Cole–Cole,stretched exponential) follow directly. The single-τiv caseis the falsifiable baseline.Separation of timescales.— Equation (3) is valid inthe regime τRC ≪ τiv, where the charge relaxationtime τRC = RchCox is much shorter than the inter-valley relaxation time. Under this assumption, the to-tal hole density p(t) = p[VGS(t)] tracks VGS instanta-neously, while fΓ(t) lags. The source and drain contactsare assumed to act as thermal reservoirs that define thecarrier injection distribution at the channel boundaries,while transport within the channel is treated in the low-bias diffusive limit [35, 36]. The channel hole densityp(VGS) = Cox(VGS −Vth)/q, with Vth the threshold volt-age, responds electrostatically to the gate. The interval-ley fraction fΓ(t) modulates the effective channel mobil-ityµeff = µK(1− fΓ) + µΓfΓ = µK −∆µ fΓ, (4)but leaves p unchanged by intervalley redistribution.This distinguishes the present mechanism from charge-trapping memtransistors, in which the trapped chargerenormalizes the channel band edge and gate effi-ciency [35]. In the present model, no additional inter-face capacitance or band-edge shift is introduced. Thedynamic drain current is thenID(t) =WLµeff [fΓ(t)] qp[VGS(t)]VDS . (5)Figure 2 illustrates the physical picture: a gate-voltagestep (VGS : V1 → V2) causes the charge to jump instan-taneously while fΓ exponentially relaxes toward the newequilibrium, producing a two-stage current response.Table I lists the numerical parameters used throughoutthis work. The effective masses and K–Γ splittings aretaken from first-principles calculations [11, 13], and themobilities are representative experimental values [6, 7].The intervalley relaxation time τiv enters only as a freeparameter; no specific numerical value is assumed in anyfigure.III. FREQUENCY-DEPENDENTTRANSCONDUCTANCEFor a small ac modulation VGS(t) = V0 + δV eiωt, lin-earizing Eq. (3) gives the dynamic valley susceptibilityχv(ω;V0) ≡δfΓδV=χv(V0)1 + iωτiv, (6)a Lorentzian whose pole is at ωc = τ−1iv . Substitutinginto the linearized current of Eq. (5):gm(ω) = gm,0(V0) +g0m,v(V0)1 + iωτiv. (7)spectroscopywindowvalley-frozen(a)(b) 0.60.00.3-0.3-0.6adiabaticFIG. 3. Frequency-dependent transconductance for bilayer(solid, blue) and trilayer (dashed, red) WSe2 at T = 300K,evaluated at the gate voltage where |g0m,v| is maximum.Shaded regions mark three dynamical regimes: adiabatic(ωτiv ≪ 1), spectroscopy window (ωτiv ∼ 1, yellow), andvalley-frozen (ωτiv ≫ 1). (a) Real part Re[gm(ω)]/g(0)m nor-malized to the dc value; dotted lines are the high-frequencylimits gm,0/g(0)m . (b) Imaginary part Im[gm(ω)]/|g0m,v|; theextremum at ωc = τ−1iv (vertical dotted) has opposite sign forbilayer (g0m,v < 0) and trilayer (g0m,v > 0).The charge-accumulation term gm,0 is frequency-independent (instantaneous charge relaxation); all dis-persion is carried by the valley term. Equation (7) is thecentral result of this paper.Separating real and imaginary parts:Re[gm(ω)] = gm,0 +g0m,v1 + (ωτiv)2, (8)Im[gm(ω)] = −g0m,v ωτiv1 + (ωτiv)2. (9)Three limiting behaviors follow immediately: (i) atωτiv ≪ 1, Re[gm] = g(0)m recovers the equilibriumanomaly of Ref. 15; (ii) at ωτiv ≫ 1, Re[gm] → gm,0,the anomaly is completely suppressed; (iii) at ω = ωc,|Im[gm]| reaches its maximum of |g0m,v|/2, providing asharp spectroscopic feature that directly identifies τiv.Because g0m,v < 0 for the bilayer and g0m,v > 0 for thetrilayer, the imaginary part peaks with opposite signs forthe two systems, providing an additional layer-numbersignature.These Kramers–Kronig-related dispersive and absorp-tive components are shown in Figs. 3(a) and 3(b), re-spectively. The high-frequency limit gm,0/g(0)m exceeds4TABLE I. Parameters used in the two-valley model.Quantity Value Descriptionm∗K 0.40m0 K-valley effective massm∗Γ 1.00m0 Γ-valley effective mass∆KΓ(2L) +26meV K–Γ valley splitting, bilayer∆KΓ(3L) −49meV K–Γ valley splitting, trilayerµK 100 cm2/(V·s) K-valley hole mobilityµΓ 30 cm2/(V·s) Γ-valley hole mobilityCox 49mF/m2 Gate oxide capacitance per unit areaaW = L 1µm Channel width and lengthVDS 50mV Drain-source bias voltageT 300K Temperaturea Equivalent oxide thickness 0.7 nm.unity for the bilayer (the valley suppression disappears)and falls below unity for the trilayer (the valley enhance-ment disappears), directly reflecting the sign of g0m,v[Fig. 3(a)]. A frequency sweep at fixed VGS = V ∗GS(the peak-|g0m,v| voltage) yields ωc from either the −3 dBrolloff of Re[gm]−gm,0 or the peak of |Im[gm]|, with bothcriteria giving the same τiv [Fig. 3(b)].IV. TRANSIENT RESPONSE ANDHYSTERESISStep response.— For a step VGS : Vi → Vf at t = 0, thecharge density jumps to p(Vf ) instantaneously while fΓsatisfies Eq. (3) with initial condition fΓ(0−) = f eqΓ (Vi) ≡fi. The solution is fΓ(t) = ff + (fi − ff )e−t/τiv withff ≡ f eqΓ (Vf ), givingID(t) = ID(∞) + ∆ItrD e−t/τiv , (10)where ID(∞) is the equilibrium current at Vf and thetransient amplitude is∆ItrDID(∞)=∆µµeff(Vf )(ff − fi). (11)The sign of ∆ItrD is governed by the layer number: forbilayer WSe2, a positive gate step drives holes fromthe light K toward the heavy Γ valley (ff > fi), so∆ItrD > 0 and the current overshoots before decaying to-ward equilibrium. For trilayer, the reverse transfer givesff < fi, ∆ItrD < 0, and the current undershoots be-fore recovering toward equilibrium. This layer-dependentdirectionality is a direct consequence of the sign rulesgn(g0m,v) = −sgn(∆KΓ) and constitutes a characteris-tic experimental fingerprint [Fig. 4(a)]. At the optimalgate voltage near the χv peak, the transient amplitudecan reach several percent of ID(∞), growing linearly with∆VGS for small steps.Hysteresis under finite-rate gate sweeps.— For a lin-early varying gate voltage at rate r = dVGS/dt, the re-laxation equation (3) gives, to first order in rτiv,fΓ(t) ≈ f eqΓ [VGS(t)]− rτiv χv[VGS(t)]. (12)The valley fraction lags the equilibrium value by δfΓ =−rτivχv. Substituting into Eq. (5), the first-order currentdeviation isδID(VGS) =WLqpVDS ∆µ τiv r χv(VGS). (13)This approximation is valid for r̃ ≡ rτiv ≪ 1 on thegate-voltage scale over which χv varies significantly. Thehysteresis current—the difference between forward (r >0) and reverse (r < 0) sweeps at the same VGS—is∆IhysD (VGS) = 2WLqpVDS ∆µ τiv |r|χv(VGS). (14)Equation (14) encodes several signatures that distin-guish valley-origin hysteresis from trap-induced effects:(i) Profile. ∆IhysD ∝ qp χv(VGS) activates only abovethreshold (where χv ̸= 0) and tracks the gate-voltageshape of the valley susceptibility. Trap hysteresis followsthe trap energy distribution and need not correlate withχv(VGS).(ii) Layer-dependent transconductance sign re-versal. ∆IhysD is positive for the bilayer (χv > 0) andnegative for the trilayer (χv < 0). Such a layer-dependenttransconductance sign reversal would be difficult to re-produce by conventional extrinsic mechanisms withoutstructural differences between the two devices.(iii) Linear sweep-rate dependence. Valley hys-teresis grows as |r|, enabling extraction ofτiv =|∆IhysD |2|r| (W/L)qp VDS∆µ |χv|(15)from the slope of peak |∆IhysD | vs |r|, using χv obtainedindependently from a static gm–VGS measurement [15] orfrom a first-principles estimate [11, 13]. Trap hysteresis5overshootundershoot(a) (b) (c)FIG. 4. Low-frequency fingerprints of delayed intervalley relaxation. (a) Signed transient amplitude ∆ItrD/ID(∞) vs gate-step size ∆VGS at T = 300K, evaluated at the gate voltage of maximum |χv|. Bilayer (blue) overshoots and trilayer (red)undershoots, encoding sgn(g0m,v); amplitudes increase linearly before saturating. The dotted line marks the ∆VGS = 0.2Vstep of Fig. 2. (b) Hysteresis current ∆IhysD (VGS) [Eq. (14)] for bilayer (solid, blue family) and trilayer (dashed, red family) atnormalized sweep rates r̃ = rτiv = 0.05, 0.15, 0.30 (labeled). Opposite signs reflect sgn(χv) = sgn(∆KΓ); grey dotted envelopestrace the normalized shape of χv(VGS), confirming the profile correspondence. (c) Peak |∆IhysD | vs r̃. Linear slopes confirmEq. (14) and give τiv directly via Eq. (15).typically follows a logarithmic or power-law dependenceon sweep rate [37–39].(iv) Subthreshold swing invariance. ∆IhysD = 0 inthe subthreshold regime where χv = 0 [15]. Trap-inducedhysteresis is often largest just below threshold where trapfilling is most active.Experimentally, the finite-rate sweep protocol corre-sponds to a periodic triangular gate-voltage waveform,as commonly employed in cyclic gate-sweep measure-ments [39]. After initial transients of order τiv follow-ing the onset of the waveform, fΓ(t) settles into a repro-ducible steady-state cycle synchronized to the triangularwaveform. In this steady-state limit, each upward anddownward branch is locally described by Eq. (12), andEq. (14) directly characterizes the steady-state hystere-sis loop width. This steady-state interpretation rendersτiv extractable from the slope of peak |∆IhysD | versus |r|without any dependence on initial conditions.Figures 4(b,c) show the predicted steady-state hystere-sis profiles ∆IhysD (VGS) for bilayer and trilayer WSe2 atthree normalized sweep rates r̃ = rτiv, together with thelinear scaling of the peak amplitude. The grey envelopetraces the normalized shape of χv(VGS), confirming thepredicted correspondence.V. TEMPERATURE DEPENDENCE ANDSPECTROSCOPY WINDOWAll three signatures are amplified at reduced tempera-ture, and the spectroscopy window shifts into the rangeof standard laboratory instruments. The valley suscep-tibility χv grows toward the intrinsic bound (4kBT )−1as T decreases [15], while phonon-mediated intervalleyscattering slows asτiv(T ) = τ0iv[eℏωph/kBT − 1](16)for a zone-edge phonon with ℏωph ≈ 25meV in WSe2 [20,40]. This Bose–Einstein form arises because phonon ab-sorption saturates at low T , leaving stimulated emissionas the rate-limiting step [22, 23]. With τ0iv ≈ 60 ps, thismodel gives τiv(100K) ≈ 1 ns, placing the spectroscopicpeak at ωc/2π ≈ 160MHz—within the bandwidth ofcommercial lock-in amplifiers and vector network analyz-ers. Below 50K, ωc/2π drops below 30MHz, and remainsaccessible with standard instrumentation. For compar-ison, ultrafast optical spectroscopy reports τiv ∼ 0.1–1 ps at room temperature in WSe2 [20, 21, 25], plac-ing ωc/2π in the 0.1–10THz range, far above standardelectrical bandwidths. The present electrical method be-comes accessible when phonon-mediated intervalley scat-tering slows substantially at reduced temperature, shift-ing ωc into the MHz–GHz window mapped in Fig. 5(a),where the shaded band marks the range 10MHz–10GHz.An Arrhenius fit of ln τiv vs 1/T within this windowyields the phonon coupling energy directly:d ln τivd(1/T )kBT≪ℏωph−−−−−−−→ ℏωphkB, (17)providing a purely electrical determination of the zone-edge phonon coupling energy [Fig. 5(b)]. The parametersτ0iv and ℏωph can be extracted from a two-parameter fitto the measured ωc(T ) curve without any optical access.6FIG. 5. Transconductance spectroscopy window and Arrhe-nius analysis. (a) |Im[gm(f)]|/|g0m,v| vs frequency at T = 50–300K (blue to red), computed using Eq. (16) with ℏωph =25meV and τ0iv = 60ps. Triangles mark the characteristicfrequency ωc/2π; the shaded band is the accessible window10MHz–10GHz. (b) Arrhenius plot of τiv vs 1000/T ; theslope of the low-temperature regime yields ℏωph/kB from apurely electrical measurement [Eq. (17)]. Colored dots corre-spond to the temperatures in (a).VI. DISCUSSIONThese three phenomena provide complementary routesto determine τiv experimentally. In ac measurements, thefrequency sweep at fixed V ∗GS gives τiv from the peak of|Im[gm(ω)]| or the −3 dB rolloff of Re[gm(ω)]−gm,0, bothpointing to the same ωc. In transient measurements, thesingle-exponential decay of ID(t)− ID(∞) directly yieldsτiv. In gate-sweep hysteresis measurements, the linear-in-|r| slope via Eq. (15) requires only a static gm–VGS curveand a rate-dependent cyclic sweep, with no rf electronics.The quasi-equilibrium limit of Ref. 15 corresponds toτiv ≪ τtr, where τtr = L2/(µKVDS) is the carrier transittime in the diffusive transport regime. For W = L =1µm, VDS = 50mV, and µK = 100 cm2 V−1 s−1 [40, 41],the transit frequency is τ−1tr ≈ 80MHz. If τiv ∼ τtr, thecrossover in Re[gm(ω)] falls in the 10–100MHz range, ac-cessible with commercial impedance analyzers. Shorter-channel devices reduce τtr. In the ballistic limit τtr isreplaced by L/vinj (with vinj the thermal injection veloc-ity), further shifting ωc to higher frequencies and provid-ing an additional experimental degree of freedom throughchannel-length scaling of the anomaly. Since τiv is setby electron–phonon coupling rather than by the chan-nel geometry, it is unchanged by device scaling. Ac-cordingly, all three normalized observables (∆ItrD/ID(∞),|Im[gm]|/|g0m,v|, and ∆IhysD /ID) are geometry-invariant.Reducing L at fixed W/L further relaxes the τRC ≪ τivcondition, because τRC = RchCox ∝ L2, making sub-micron devices favorable for the measurement. At highfrequencies, careful rf de-embedding of contact resistanceand pad parasitics will be required to isolate the intrinsicgm(ω) from extrinsic contributions. Although the dis-cussion is illustrated for a dual-gate geometry, the mech-anism is not specific to a particular gate architecture:gate-all-around devices [14, 42] enhance the electrostaticcoupling and hence the magnitude of pχv, whereas dual-gate devices offer a more readily accessible experimentalplatform.Several other mechanisms can produce frequency-dependent gm or hysteresis and must be distinguished.Interface states and fast charge exchange are known todegrade the subthreshold swing, whereas border trapsproduce hysteresis and threshold-voltage drifts [38, 43].By contrast, the valley hysteresis predicted here arisesfrom delayed intervalley redistribution and does not in-troduce an additional interface-trap capacitance; withinthe present model, the subthreshold swing remains un-changed. Charge trapping in dielectrics can produce τvalues from microseconds to seconds, but shows no layer-dependent transconductance sign reversal and no depen-dence on ∆KΓ. Applying compressive biaxial strain,which shifts ∆KΓ predictably [14, 44], would modifythe valley contribution in a controlled manner, pro-viding a decisive test. Deviations from the predictedLorentzian line shape of Im[gm(ω)] would directly sig-nal distributed intervalley relaxation channels beyond theminimal single-τiv model, opening a route to resolving therelaxation spectrum electrically.VII. CONCLUSIONIn summary, we have shown that the valley popula-tion in multilayer WSe2 field-effect transistors acts asa dynamic internal state variable with relaxation timeτiv. The resulting frequency-dependent transconduc-tance gm(ω) = gm,0 + g0m,v/(1 + iωτiv), two-stage tran-sient response, and sweep-rate-proportional hysteresiseach provide distinct and electrically accessible probesof intervalley relaxation dynamics. The layer-dependenttransconductance sign reversal and gate-voltage profile ofall three observables constitute characteristic fingerprintsof valley-thermodynamic origin, completing the dynamic7extension of the valley-thermodynamics framework fortwo-dimensional transistors [15].ACKNOWLEDGMENTSThis work was supported by JSPS KAKENHI(Grants No. JP25K01609, No. JP22H05473, andNo. JP21H01019), JST CREST (Grant No. JP-MJCR19T1). K.W. acknowledges financial support fromthe Sumitomo Foundation (Grant No. 2401203).DATA AVAILABILITYThe data used and analyzed during the current studyare available from the corresponding author upon reason-able request.[1] Q. H. Wang, K. Kalantar-Zadeh, A. Kis, J. N. Coleman,and M. S. Strano, Electronics and optoelectronics of two-dimensional transition metal dichalcogenides, Nat. Nan-otechnol. 7, 699 (2012).[2] S. Manzeli, D. Ovchinnikov, D. Pasquier, O. V. Yazyev,and A. Kis, 2D transition metal dichalcogenides, Nat.Rev. Mater. 2, 17033 (2017).[3] K. F. Mak, D. Xiao, and J. Shan, Light–valley interac-tions in 2d semiconductors, Nat. Photon. 12, 451 (2018).[4] J. R. Schaibley, H. Yu, G. Clark, P. Rivera, J. S. Ross,K. L. Seyler, W. Yao, and X. Xu, Valleytronics in 2dmaterials, Nat. Rev. Mater. 1, 16055 (2016).[5] B. Radisavljevic, A. Radenovic, J. Brivio, V. Giacometti,and A. Kis, Single-layer MoS2 transistors, Nat. Nanotech-nol. 6, 147 (2011).[6] H. Fang, S. Chuang, T. C. Chang, K. Takei, T. Taka-hashi, and A. Javey, High-performance single layeredWSe2 p-FETs with chemically doped contacts, NanoLett. 12, 3788 (2012).[7] H. C. P. Movva, A. Rai, S. Kang, K. Kim, B. Fallahazad,T. Taniguchi, K. Watanabe, E. Tutuc, and S. K. Baner-jee, High-mobility holes in dual-gated WSe2 field-effecttransistors, ACS Nano 9, 10402 (2015).[8] I. Kim, N. Higashitarumizu, I. K. M. R. Rahman,S. Wang, H. M. Kim, J. Geng, R. R. Prabhakar, J. W.Ager, and A. Javey, Low contact resistance WSe2 p-type transistors with highly stable, CMOS-compatibledopants, Nano Lett. 24, 13528 (2024).[9] W. Zhao, Z. Ghorannevis, L. Chu, M. Toh, C. Kloc, P.-H. Tan, and G. Eda, Evolution of electronic structure inatomically thin sheets of WS2 and WSe2, ACS Nano 7,791 (2013).[10] Y. Zhang, T.-R. Chang, B. Zhou, Y.-T. Cui, H. Yan,Z. Liu, F. Schmitt, J. Lee, R. Moore, Y. Chen, H. Lin,H.-T. Jeng, S.-K. Mo, Z. Hussain, A. Bansil, and Z.-X.Shen, Direct observation of the transition from indirect todirect bandgap in atomically thin epitaxial MoSe2, Nat.Nanotechnol. 9, 111 (2014).[11] D. Wickramaratne, F. Zahid, and R. K. Lake, Elec-tronic and thermoelectric properties of few-layer transi-tion metal dichalcogenides, J. Chem. Phys. 140, 124710(2014).[12] Z. Zhu, Y. Cheng, and U. Schwingenschlögl, Gi-ant spin-orbit-induced spin splitting in two-dimensionaltransition-metal dichalcogenide semiconductors, Phys.Rev. B 84, 153402 (2011).[13] G.-B. Liu, W.-Y. Shan, Y. Yao, W. Yao, and D. Xiao,Three-band tight-binding model for monolayers of group-VIB transition metal dichalcogenides, Phys. Rev. B 88,085433 (2013).[14] K. Wakabayashi, S. Adhikary, and K. Tsukagoshi,Valley-landscape engineering in bilayer WSe2 gate-all-around transistors, Phys. Rev. Appl. (2026), submitted,arXiv:2606.08955 [cond-mat.mes-hall].[15] K. Wakabayashi, S. Adhikary, and T. Kameda, Transcon-ductance as a probe of valley thermodynamics inmultilayer WSe2, Phys. Rev. B 114, 115403 (2026),arXiv:2605.19212 [cond-mat.mes-hall].[16] D. Xiao, G.-B. Liu, W. Feng, X. Xu, and W. Yao, Cou-pled spin and valley physics in monolayers of MoS2 andother group-VI dichalcogenides, Phys. Rev. Lett. 108,196802 (2012).[17] K. F. Mak, K. He, J. Shan, and T. F. Heinz, Control ofvalley polarization in monolayer MoS2 by optical helicity,Nat. Nanotechnol. 7, 494 (2012).[18] H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, Valley po-larization in MoS2 monolayers by optical pumping, Nat.Nanotechnol. 7, 490 (2012).[19] K. F. Mak, K. L. McGill, J. Park, and P. L. McEuen,The valley Hall effect in MoS2 transistors, Science 344,1489 (2014).[20] D. H. Lee, S.-J. Choi, H. Kim, Y.-S. Kim, and S. Jung,Direct probing of phonon mode specific electron–phononscatterings in two-dimensional semiconductor transitionmetal dichalcogenides, Nat. Commun. 12, 4520 (2021).[21] S. Bae, K. Matsumoto, H. Raebiger, K.-i. Shudo, Y.-H.Kim, Ø. S. Handeg̊ard, T. Nagao, M. Kitajima, Y. Sakai,X. Zhang, R. Vajtai, P. Ajayan, J. Kono, J. Takeda, andI. Katayama, K-point longitudinal acoustic phonons areresponsible for ultrafast intervalley scattering in mono-layer MoSe2, Nat. Commun. 13, 4279 (2022).[22] G. Kioseoglou, A. T. Hanbicki, M. Currie, A. L. Fried-man, D. Gunlycke, and B. T. Jonker, Valley polariza-tion and intervalley scattering in monolayer MoS2, Appl.Phys. Lett. 101, 221907 (2012).[23] R. Schmidt, G. Berghäuser, R. Schneider, M. Selig,P. Tonndorf, E. Malić, A. Knorr, S. Michaelis de Vascon-cellos, and R. Bratschitsch, Ultrafast Coulomb-inducedintervalley coupling in atomically thin WS2, Nano Lett.16, 2945 (2016).[24] S. Dal Conte, F. Bottegoni, E. A. A. Pogna, D. De Fazio,S. Ambrogio, I. Bargigia, C. D’Andrea, A. Lombardo,M. Bruna, F. Ciccacci, A. C. Ferrari, G. Cerullo, andM. Finazzi, Ultrafast valley relaxation dynamics in mono-layer MoS2 probed by nonequilibrium optical techniques,Phys. Rev. B 92, 235425 (2015).[25] J. Oh, H.-T. Chang, C. T. Chen, S. Aloni,A. Schwartzberg, and S. R. Leone, Carrier and phonondynamics in multilayer WSe2 captured by extreme ultra-violet transient absorption spectroscopy, J. Phys. Chem.https://doi.org/10.1038/nnano.2012.193https://doi.org/10.1038/nnano.2012.193https://doi.org/10.1038/natrevmats.2017.33https://doi.org/10.1038/natrevmats.2017.33https://doi.org/10.1038/s41566-018-0204-6https://doi.org/10.1038/natrevmats.2016.55https://doi.org/10.1038/nnano.2010.279https://doi.org/10.1038/nnano.2010.279https://doi.org/10.1021/nl301702rhttps://doi.org/10.1021/nl301702rhttps://doi.org/10.1021/acsnano.5b04611https://doi.org/10.1021/acs.nanolett.4c02948https://doi.org/10.1021/nn305275hhttps://doi.org/10.1021/nn305275hhttps://doi.org/10.1038/nnano.2013.277https://doi.org/10.1038/nnano.2013.277https://doi.org/10.1063/1.4869142https://doi.org/10.1063/1.4869142https://doi.org/10.1103/PhysRevB.84.153402https://doi.org/10.1103/PhysRevB.84.153402https://doi.org/10.1103/PhysRevB.88.085433https://doi.org/10.1103/PhysRevB.88.085433https://arxiv.org/abs/2606.08955https://doi.org/10.1103/xsr9-l7llhttps://arxiv.org/abs/2605.19212https://doi.org/10.1103/PhysRevLett.108.196802https://doi.org/10.1103/PhysRevLett.108.196802https://doi.org/10.1038/nnano.2012.96https://doi.org/10.1038/nnano.2012.95https://doi.org/10.1038/nnano.2012.95https://doi.org/10.1126/science.1250140https://doi.org/10.1126/science.1250140https://doi.org/10.1038/s41467-021-24875-2https://doi.org/10.1038/s41467-022-32008-6https://doi.org/10.1063/1.4768299https://doi.org/10.1063/1.4768299https://doi.org/10.1021/acs.nanolett.5b04733https://doi.org/10.1021/acs.nanolett.5b04733https://doi.org/10.1103/PhysRevB.92.235425https://doi.org/10.1021/acs.jpcc.2c076958C 127, 5004 (2023).[26] O. Dogadov, H. Mittenzwey, M. Bertolotti, N. Olsen,T. Deckert, C. Trovatello, X. Zhu, D. Brida, G. Cerullo,A. Knorr, and S. Dal Conte, Dissecting intervalley cou-pling mechanisms in monolayer transition metal dichalco-genides, npj 2D Mater. Appl. 10, 21 (2026).[27] G. Wang, X. Marie, B. L. Liu, T. Amand, C. Robert,F. Cadiz, P. Renucci, and B. Urbaszek, Control of exci-ton valley coherence in transition metal dichalcogenidemonolayers, Phys. Rev. Lett. 117, 187401 (2016).[28] R. Bertoni, C. W. Nicholson, L. Waldecker, H. Hübener,C. Monney, U. De Giovannini, M. Puppin, M. Hoesch,E. Springate, R. T. Chapman, C. Cacho, M. Wolf, A. Ru-bio, and R. Ernstorfer, Generation and evolution of spin-,valley-, and layer-polarized excited carriers in inversion-symmetric WSe2, Phys. Rev. Lett. 117, 277201 (2016).[29] M. M. Glazov, T. Amand, X. Marie, D. Lagarde,L. Bouet, and B. Urbaszek, Exciton fine structureand spin decoherence in monolayers of transition metaldichalcogenides, Phys. Rev. B 89, 201302 (2014).[30] J. Madéo, M. K. L. Man, C. Sahoo, M. Campbell, V. Pa-reek, E. L. Wong, A. Al-Mahboob, N. S. Chan, A. Kar-makar, B. M. K. Mariserla, X. Li, T. F. Heinz, T. Cao,and K. M. Dani, Directly visualizing the momentum-forbidden dark excitons and their dynamics in atomicallythin semiconductors, Science 370, 1199 (2020).[31] R. Wallauer, R. Perea-Cauśın, L. Münster, S. Zajusch,S. Brem, J. Güdde, K. Tanimura, K.-Q. Lin, R. Huber,E. Malic, and U. Höfer, Momentum-resolved observationof exciton formation dynamics in monolayer WS2, NanoLett. 21, 5867 (2021).[32] A. K. Geim and I. V. Grigorieva, Van der Waals het-erostructures, Nature 499, 419 (2013).[33] C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang,S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L.Shepard, and J. Hone, Boron nitride substrates for high-quality graphene electronics, Nat. Nanotechnol. 5, 722(2010).[34] K. S. Cole and R. H. Cole, Dispersion and absorption indielectrics I. alternating current characteristics, J. Chem.Phys. 9, 341 (1941).[35] R. S. W. Silva, S. Pradhan, F. Hartmann, L. K. Caste-lano, O. Lipan, S. Höfling, and V. Lopez-Richard, Over-coming the Boltzmann limit in two-dimensional mem-transistors via hysteretic charge trapping, Phys. Rev.Appl. 25, L051004 (2026).[36] D. Jiménez, Drift-diffusion model for single layer tran-sition metal dichalcogenide field-effect transistors, Appl.Phys. Lett. 101, 243501 (2012).[37] S. Ghatak, A. N. Pal, and A. Ghosh, Nature of elec-tronic states in atomically thin MoS2 field-effect transis-tors, ACS Nano 5, 7707 (2011).[38] Y. Y. Illarionov, T. Knobloch, M. Jech, M. Lanza,D. Akinwande, M. I. Vexler, T. Mueller, M. C. Lemme,G. Fiori, F. Schwierz, and T. Grasser, Insulators for 2dnanoelectronics: The gap to bridge, Nat. Commun. 11,3385 (2020).[39] A. Karl, A. Verdianu, D. Waldhoer, T. Knobloch,J. Kurzweil, M. Bahrami, M. R. Davoudi, P. Khak-baz, B. Stampfer, S. M. Sattari-Esfahlan, Y. Illarionov,A. Nazir, C. Liu, Y. Zheng, L. Pettorosso, D. Polyushkin,T. Müller, S. Das, X. R. Wang, J. Tang, Y. Zhang,C. Tan, Y. Li, H. Peng, M. Waltl, and T. Grasser,A standardized approach to characterize hysteresis in2d-materials-based transistors for stability benchmark-ing and performance projection, Nat. Commun. 17, 171(2026).[40] Z. Jin, X. Li, J. T. Mullen, and K. W. Kim, Intrin-sic transport properties of electrons and holes in mono-layer transition-metal dichalcogenides, Phys. Rev. B 90,045422 (2014).[41] K. Kaasbjerg, K. S. Thygesen, and K. W. Jacobsen,Phonon-limited mobility in n-type single-layer MoS2from first principles, Phys. Rev. B 85, 115317 (2012).[42] S. Mukesh and J. Zhang, A review of the gate-all-aroundnanosheet FET process opportunities, Electronics 11,3589 (2022).[43] T. Knobloch, G. Rzepa, Y. Y. Illarionov, M. Waltl,F. Schanovsky, B. Stampfer, M. M. Furchi, T. Mueller,and T. Grasser, A physical model for the hysteresis inMoS2 transistors, IEEE J. Electron Devices Soc. 6, 972(2018).[44] P. Johari and V. B. Shenoy, Tuning the electronic prop-erties of semiconducting transition metal dichalcogenidesby applying mechanical strains, ACS Nano 6, 5449(2012).https://doi.org/10.1021/acs.jpcc.2c07695https://doi.org/10.1038/s41699-025-00653-2https://doi.org/10.1103/PhysRevLett.117.187401https://doi.org/10.1103/PhysRevLett.117.277201https://doi.org/10.1103/PhysRevB.89.201302https://doi.org/10.1126/science.aba1029https://doi.org/10.1021/acs.nanolett.1c01839https://doi.org/10.1021/acs.nanolett.1c01839https://doi.org/10.1038/nature12385https://doi.org/10.1038/nnano.2010.172https://doi.org/10.1038/nnano.2010.172https://doi.org/10.1063/1.1750906https://doi.org/10.1063/1.1750906https://doi.org/10.1103/3m8n-ctrvhttps://doi.org/10.1103/3m8n-ctrvhttps://doi.org/10.1063/1.4769587https://doi.org/10.1063/1.4769587https://doi.org/10.1021/nn202852jhttps://doi.org/10.1038/s41467-020-16640-8https://doi.org/10.1038/s41467-020-16640-8https://doi.org/10.1038/s41467-025-66210-zhttps://doi.org/10.1038/s41467-025-66210-zhttps://doi.org/10.1103/PhysRevB.90.045422https://doi.org/10.1103/PhysRevB.90.045422https://doi.org/10.1103/PhysRevB.85.115317https://doi.org/10.3390/electronics11213589https://doi.org/10.3390/electronics11213589https://doi.org/10.1109/JEDS.2018.2829933https://doi.org/10.1109/JEDS.2018.2829933https://doi.org/10.1021/nn301320rhttps://doi.org/10.1021/nn301320r Electrical Spectroscopy of Intervalley Relaxation in WSe2 Transistors Abstract Introduction Dynamic Valley Model Frequency-Dependent Transconductance Transient Response and Hysteresis Temperature Dependence and Spectroscopy Window Discussion Conclusion Acknowledgments Data Availability References