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[Omid Ghaebi](https://orcid.org/0009-0003-0290-0723), [Sebastian Klimmer](https://orcid.org/0000-0002-8582-3588), [Nele Tornow](https://orcid.org/0009-0006-3921-8981), [Niels Buijssen](https://orcid.org/0009-0006-9984-0027), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Andrea Tomadin](https://orcid.org/0000-0001-9117-9891), [Habib Rostami](https://orcid.org/0000-0002-9521-1008), [Giancarlo Soavi](https://orcid.org/0000-0003-2434-2251)

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[Ultrafast Opto‐Electronic and Thermal Tuning of Third‐Harmonic Generation in a Graphene Field Effect Transistor](https://mdr.nims.go.jp/datasets/d538cd59-a9f7-4e68-9e00-d58f117d12cf)

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Ultrafast Opto‐Electronic and Thermal Tuning of Third‐Harmonic Generation in a Graphene Field Effect TransistorRESEARCH ARTICLEwww.advancedscience.comUltrafast Opto-Electronic and Thermal Tuning ofThird-Harmonic Generation in a Graphene Field EffectTransistorOmid Ghaebi, Sebastian Klimmer, Nele Tornow, Niels Buijssen, Takashi Taniguchi,Kenji Watanabe, Andrea Tomadin, Habib Rostami, and Giancarlo Soavi*Graphene is a unique platform for tunable opto-electronic applications thanksto its linear band dispersion, which allows electrical control of resonantlight-matter interactions. Tuning the nonlinear optical response of graphene ispossible both electrically and in an all-optical fashion, but each approachinvolves a trade-off between speed and modulation depth. Here, latticetemperature, electron doping, and all-optical tuning of third-harmonicgeneration are combined in a hexagonal boron nitride-encapsulatedgraphene opto-electronic device and demonstrate up to 85% modulationdepth along with gate-tunable ultrafast dynamics. These results arisefrom the dynamic changes in the transient electronic temperaturecombined with Pauli blocking induced by the out-of-equilibrium chemicalpotential. The work provides a detailed description of the transient nonlinearoptical and electronic response of graphene, which is crucial for the design ofnanoscale and ultrafast optical modulators, detectors, and frequencyconverters.1. Introduction2D materials are ideal candidates for nonlinear optical ap-plications at the nanoscale,[1] as they enable ultra-broadbandO. Ghaebi, S. Klimmer, N. Tornow, N. Buijssen, G. SoaviInstitute of Solid State PhysicsFriedrich Schiller University Jena07743 Jena, GermanyE-mail: giancarlo.soavi@uni-jena.deS. KlimmerARC Centre of Excellence for Transformative Meta-Optical SystemsDepartment of Electronic Materials EngineeringResearch School of PhysicsThe Australian National UniversityCanberra ACT 2601, AustraliaThe ORCID identification number(s) for the author(s) of this articlecan be found under https://doi.org/10.1002/advs.202401840© 2024 The Author(s). Advanced Science published by Wiley-VCHGmbH. This is an open access article under the terms of the CreativeCommons Attribution License, which permits use, distribution andreproduction in any medium, provided the original work is properly cited.DOI: 10.1002/advs.202401840optical parametric amplification,[2] sponta-neous parametric down-conversion,[3] elec-trical, and all-optical tuning of the sec-ond harmonic (SH)[4–7] and third harmonic(TH) generation,[5,8,9] giant efficiencies ofTHz high harmonic generation,[10] andapplications in integrated nonlinear opto-electronic devices such as gas sensors,[11]logic gates,[12,13] and valleytronics.[14,15]Text within the family of 2D materi-als, graphene arguably shows the most in-triguing nonlinear response. Being cen-trosymmetric, the first nonlinear term inits polarization is the third-order suscep-tibility 𝜒 (3). While few experimental stud-ies have observed second-harmonic gen-eration (SHG) due to breaking of sym-metry at an interface,[16,17] in-plane elec-tric fields and currents[18,19] or from theelectric quadrupole response.[20] the vastmajority of nonlinear optical experimentson graphene have focused on 𝜒 (3) processes such as four-wavemixing (FWM),[21] third-harmonic generation (THG),[8,9,22–24]and saturable absorption.[25–27] In particular, THG and FWMhave recently gained increasing attention following theT. TaniguchiResearch Center for Materials NanoarchitectonicsNational Institute for Materials Science1-1 Namiki, Tsukuba 305-0044, JapanK. WatanabeResearch Center for Electronic and Optical MaterialsNational Institute for Materials Science1-1 Namiki, Tsukuba 305-0044, JapanA. TomadinDipartimento di FisicaUniversità di PisaLargo Bruno Pontecorvo 3, Pisa 56127, ItalyH. RostamiDepartment of PhysicsUniversity of BathClaverton Down, Bath BA2 7AY, UKG. SoaviAbbe Center of PhotonicsFriedrich Schiller University Jena07743 Jena, GermanyAdv. Sci. 2024, 11, 2401840 2401840 (1 of 9) © 2024 The Author(s). Advanced Science published by Wiley-VCH GmbHhttp://www.advancedscience.commailto:giancarlo.soavi@uni-jena.dehttps://doi.org/10.1002/advs.202401840http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/http://crossmark.crossref.org/dialog/?doi=10.1002%2Fadvs.202401840&domain=pdf&date_stamp=2024-06-18www.advancedsciencenews.com www.advancedscience.comFigure 1. Opto-electronic modulation of THG in a graphene FET. a) Sketch and microscope optical image of the device. Monolayer graphene is encap-sulated between two hBN flakes. VG, VD, 𝜔FB, and 𝜔CB represent the gate-source voltage, source-drain voltage, fundamental beam, and control beam,respectively. b) THG as a function of EF (bottom x-axis) and VG (top x-axis) at lattice temperatures of TL = 295 K (red curve) and TL = 30 K (blue curve).The black curve is the drain current (ID) as a function of EF and VG at the drain voltage of VD = 1 mV.demonstration of their electrical[8,9,21] and all-optical[5] mod-ulation, which provide a route towards ultrafast nanoscalefrequency converters and a powerful method to probe ultrafasthot electron dynamics. The electrical tunability of THG ingraphene has been widely explored,[8,9,21,22] whereas ultrafastall-optical modulation and the interplay of lattice (TL) and elec-tron temperatures (Te) in high-quality hexagonal boron nitride(hBN)-encapsulated graphene samples are scarcely studied.In this work, we provide a detailed experimental and theoret-ical study of ultrafast thermal and opto-electronic modulationof THG in a high-quality and gate-tunable hBN/graphene/hBNfield effect transistor (FET). Encapsulation of graphene in hBNis widely used to achieve a higher sample quality, and to engi-neer the Te via out of plane heat transfer.[28,29] In our experi-ments, we further use hBN encapsulation to reduce the intrin-sic doping of graphene, and with this, we demonstrate for thefirst time ambipolar gate tunable THG, as discussed in the fol-lowing. Our scheme for opto-electronic THG modulation canbe briefly summarized as follows. We irradiate graphene withtwo pulses: a fundamental beam (FB) and a control beam (CB).The FB is responsible for inducing the parametric THG process(𝜔FB → 3𝜔FB) while the CB controls the TH efficiency via tun-ing of Te and Pauli blocking. We point out from the very startthat the FB affects Te and Pauli blocking as well, due to its largefluence (comparable to the CB), necessary to generate a sizableTH. Furthermore, electrical doping by means of external gatesenables the system to modulate the competition between Te andPauli blocking mechanisms and to tune the TH ultrafast recom-bination dynamics. Thus, by combining electrical and all-opticalcontrol of Te and Fermi Energy (EF), we achieve active modula-tion of THG in graphene with the following main results. First,experiments on hBN-encapsulated samples allow to show thatthe electrical modulation of THG in graphene is symmetric forelectrons and holes within the Dirac cone. This is the nonlin-ear optical analog of the electronic ambipolar behavior of FETs,which was absent in previous studies.[8,21] Further, we observeup to 300% modulation in the THG intensity by tuning TL from295 to 33 K. Second, we show that electrical doping can be usedto actively control the recombination dynamics of the TH signalarising from phase-space quenching of the scattering betweenhot electrons and optical phonons.[30] Third, we shed light onthe physical origin of the ultrafast TH modulation and the inter-play of hot electrons and Pauli blocking. Finally, with our nonlin-ear opto-electronic device, we achieve a TH modulation depth of≈85% at EF = 300 meV and peak fluence of 200 μJ cm−2, namelya two orders of magnitude enhancement in the modulation ef-ficiency (i.e., modulation depth per unit of fluence) comparedto previous reports.[5] This is possible thanks to mid-IR excita-tion and active control of EF and TL and thus it further clarifiesthat a deeper understanding of the ultrafast and nonlinear opto-electronic response of graphene is paramount for the design andoptimization of nanoscale ultrafast devices, such as optical mod-ulators, detectors, and frequency converters.2. Ambipolar Gate-Tunable THGOpto-electronic (i.e., optical and electrical) modulation of THG isperformed on a back-gated FET based on a single layer grapheneencapsulated in two ≈10 nm thick hBN layers (Figure 1a). Thedevice was prepared by mechanical exfoliation and dry transfer,following the approach described in ref. [31] (see Sections S1and S2, Supporting Information for details on sample fabricationand characterization). For the THG measurements, we used twosynchronized laser pulses at a repetition rate of 76 MHz, photonenergies of 0.32 eV (3900 nm) and 1.2 eV (1030 nm) and pulseduration of ≈150 fs/110 fs for the FB/CB, respectively. The spot-sizes of the focused FB and CB have been measured using therazor blade technique[32] and they are ≈6.7 and ≈2.2 μm, respec-tively (see Section S3, Supporting Information).First, we measured gate-tunable THG with a “static” proce-dure (i.e., without CB). We irradiate our device with the FB(130 μJ cm−2) and collect the TH power for different values ofthe applied VG in the range −30 to 30 V, corresponding to valuesof the EF in the range −300 to 300 meV (see Section S2, Support-ing Information for the calculation of EF) and for different TL. Westress that, under these experimental conditions, the TH inten-sity from the hBN encapsulant is negligible (see Section S4, Sup-porting Information). The experimental data (Figure 1b) show amodulation factor of ≈4 when TL = 295 K and the EF is tunedfrom ≈50 to 300 meV. This gate-tunable TH modulation is dueto the crossing of multi-photon resonances in the Dirac cone, aslargely discussed in refs. [8, 9]. Once the TL is decreased to 33 K,Adv. Sci. 2024, 11, 2401840 2401840 (2 of 9) © 2024 The Author(s). Advanced Science published by Wiley-VCH GmbH 21983844, 2024, 31, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/advs.202401840 by Kenji Watanabe - National Institute For , Wiley Online Library on [27/08/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensehttp://www.advancedsciencenews.comhttp://www.advancedscience.comwww.advancedsciencenews.com www.advancedscience.comthe modulation factor in the same EF range increases to ≈9. Com-paring the two curves at different temperatures, we observe anenhancement of the TH power while reducing TL of ≈1.5 and ≈3at EF = 50 meV and EF = 300 meV, respectively. The origin of thisremarkable enhancement of THG with TL is manifold. Our the-oretical analysis reproduces this effect, on a smaller magnitude,solely based on the different electron distribution achieved whensamples with different TL are irradiated by the same FB. This isan indirect effect of TL on THG, due to the different dynamics ex-perienced by electrons on a statistical level. However, we assumethat a contribution to the observed TH enhancement arises alsofrom a direct effect of temperature at the level of single-particle,coherent evolution during the FB pulse duration. Such an effectcan be attributed to the temperature-dependent electron scatter-ing rates (or electron spectral broadening) with impurities, de-fects and phonons (see Section S6, Supporting Information). Al-though our numerical calculations support this argument, a soliddetermination of the scattering rates at different temperatureswould require a much larger amount of data sets that is outsidethe scope of this work.The absence of sharp peaks in the data reported in Figure 1bis a clear indication of the high Te reached during theexperiments,[8,9] as we discuss in detail in the Section S5 (Sup-porting Information). Since Te is a function of EF and varies dra-matically over the pulse duration, we cannot assign a single valueof Te to the points in Figure 1b. However, if we consider, e.g., TL= 33 K and EF = 50 meV, our calculations show that a Te > 1400 Kis achieved by the electron distribution for over 200 fs, at the FBpeak fluence of 130 μJ cm−2 (see also Section S5, Supporting In-formation). We point out that we observe gate-tunable THG forboth positive and negative values of the EF, indicating that theTH enhancement at multi-photon resonances can be achievedfor both n- and p-doping, i.e., in the conduction and valence bandof the Dirac cone, qualitatively preserving the electron-hole sym-metry of the phenomenon to a remarkable degree.Finally, the results reported in Figure 1b allow us to estimatethe 𝜒 (3) of graphene at different values of EF, at the FB photonenergy of 0.32 eV by using the two following equations:[22]P(𝜔i,o) =18(𝜋ln 2)3∕2f 𝜏W2n𝜔i,o𝜖0c|E(𝜔i,o)|22(1)E(𝜔o) =14i𝜔i2𝜋c𝜒(3)expdgrE3(𝜔i) (2)where P(𝜔i, o), E(𝜔i, o) are the input/generated TH power and elec-tric field and f,𝜏, n𝜔i,oare the repetition rate, pulse duration, andrefractive index, respectively. The input/TH electric fields canbe extracted from Equation (1) and then the 𝜒 (3) value can becalculated using Equation (2). dgr = 0.3 nm is the thickness ofmonolayer graphene. Considering the losses of the setup (seeSection S3, Supporting Information) and TL = 33 K we obtain𝜒 (3) ≈ 2 × 10−15 m2 V−2 for EF ≈ 300 meV and ≈ 8 × 10−16 m2 V−2for EF ≈ 0 meV, in agreement with ref. [22] where a 𝜒 (3) ≈ 6 ×10−16 m2 V−2 was reported for pristine graphene at a fundamen-tal photon energy of 0.225 eV and EF = 390 meV.3. Ultrafast Opto-Electronic TH ModulationNext, we shift our attention to time-resolved and all-optical THmodulation. We initially fix the CB and FB fluence at 170 and110 μJ cm−2, respectively, and scan their relative delay for differ-ent values of EF in the range 0 to 390 meV. We remark that thisrange of EF overlaps the region defined by the lower threshold EF> ℏ𝜔/2, where absorption of the FB, at zero temperature, is for-bidden by Pauli blocking. However, we do not see an abrupt drop-off of the measured signal when the (EF) exceeds such threshold.The reason is that the finite temperature in our samples ensuresthat a residual absorption is always present. Even a small initialabsorption produces a rapid temperature increase, which broad-ens the electron distribution in the energy space and relaxes thecondition for Pauli blocking. To mitigate the effect of diminishedabsorption, in the following, we discuss the behavior of the mea-sured signal divided by the signal before the pump is applied,thus “normalizing-out” the most trivial part of the Pauli block-ing. We point out, however, that other non-trivial thresholds ap-pear in the THG as the Fermi energy crosses multiples of theFB frequency.[8,9] Figure 2 shows the experimental results for theratio ΔTHG/THG0, whereΔTHG(𝜏) = THG(𝜏) − THG0 (3)THG(𝜏) is the measured signal as a function of delay 𝜏, and THG0is the reference TH signal measured in the absence of the CB,that we measure at a negative delay 𝜏 = −2 ps. As expected, thesignal features a sharp peak when the FB overlaps with the CB,i.e., when both beams excite the electron system, followed by a“relaxation” stage converging to a zero signal, which representsthe recovery of the system from the excitation due to the CB. Atlarge delays, the effect of the CB vanishes and the TH signal re-covers to its reference value THG0.The process of electron relaxation in graphene after excitationfrom an ultrashort pulse has been discussed at length in theliterature,[33–38] and in summary, involves: i) an initial stage dom-inated by electron–electron interactions where the photoexcitedelectron system achieves thermalization at a temperature muchhigher than the initial (lattice) temperature, possibly with inter-band processes associated to Auger recombination and carriermultiplication; ii) a first cooling stage dominated by the emis-sion of optical phonons where both the electron temperatureand the photoexcited density decreases; iii) a second, slower cool-ing stage, where the hot optical phonons thermalize with theacoustic phonons of the lattice, possibly with the interventionof “supercollision” processes, and the unperturbed initial stateis finally recovered. We remark again that the FB, due to itsfluence, strongly perturbs the electron system, such that, evenseveral ps after the CB, the TH signal cannot be consideredas the response of an electron system at equilibrium with thelattice.From the data in Figure 2, we also notice that the rate of re-laxation diminishes as the Fermi energy is increased. We rec-ognize this effect as the quenching of optical phonon emissionin the first cooling stage, due to the reduction of the availablephase-space for electronic transitions, which was recently dis-cussed in ref. [30]. In other words, due to Pauli blocking, pho-toexcited electrons at energy E can only emit a phonon of energyℏ𝜔ph, if states are available at energy E − ℏ𝜔ph. As the Fermi en-ergy is increased, and approaches the photoexcitation energy, thiscondition is harder and harder to satisfy, even at large tempera-ture where the electron distribution is broadened. It is interestingthat this phase-space effect does not only affect the differentialAdv. Sci. 2024, 11, 2401840 2401840 (3 of 9) © 2024 The Author(s). Advanced Science published by Wiley-VCH GmbH 21983844, 2024, 31, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/advs.202401840 by Kenji Watanabe - National Institute For , Wiley Online Library on [27/08/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensehttp://www.advancedsciencenews.comhttp://www.advancedscience.comwww.advancedsciencenews.com www.advancedscience.comFigure 2. All-optical modulation of THG and gate tunable dynamics. a) Ratio ΔTHGTHG0, (defined in equation (3)), for different values of EF. THG0 has beenmeasured at −2 ps. The interplay of transient heating and PB (Pauli blocking) on electrons will occur when CB and FB pulses are spatially and temporallysynchronized and subsequent cooling occurs via electron–electron and electron–phonon scattering. b) Normalized ΔTHGTHG0for EF = 120 meV and EF =350 meV.transmission of the electron system, as demonstrated in ref. [30],but emerges in the measurement of the TH as well. This obser-vation highlights how consequential it is to be able to tune theelectron density by electrical doping in a graphene-based opto-electronic device, thus exerting a certain degree of control on bothits linear and non-linear optical response.Finally, we explore the dependence of THG on the state of theelectron system before the FB, by changing the fluence of the CB.In Figure 3a we plot the THG efficiency (THGE) and in Figure 3bthe third harmonic modulation depth (TH-MD), definedasTHGE =PTHPFB, TH − MD =ΔPTHPTH0(4)respectively, where ΔPTH is the difference in the TH power (PTH)with and without (PTH0) the CB, and PFB is the power of the fun-damental beam. The data are shown as a function of EF and forFigure 3. Influence of EF and CB peak fluence on THGE and TH-MD. a,b) Experimental THGE and TH-MD (defined in Equation (4)) as a function ofEF, for different values of the CB peak fluence reported in the legend. c,d) Theoretical THGE and TH-MD calculated using the experimental values of theincident peak fluences of CB and FB.Adv. Sci. 2024, 11, 2401840 2401840 (4 of 9) © 2024 The Author(s). Advanced Science published by Wiley-VCH GmbH 21983844, 2024, 31, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/advs.202401840 by Kenji Watanabe - National Institute For , Wiley Online Library on [27/08/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensehttp://www.advancedsciencenews.comhttp://www.advancedscience.comwww.advancedsciencenews.com www.advancedscience.comdifferent values of the incident CB fluence. In all the experimen-tal graphs, the data are extracted at zero time delay between theFB and CB.When the CB is off (black symbols in Figure 3a), we obtaina similar result reported in Figure 1b, namely an increase of theTHGE when ℏ𝜔< 2EF. The same trend can be observed when weswitch-on the CB, but in this case, the modulation factor with re-spect to EF is reduced. When the CB fluence reaches 200 μJ cm−2(green symbols) the modulation factor is close to zero and theTHGE is almost constant over the measured range of EF.The TH-MD is in the range ≈7 to 85% for CB peak fluencesof 11 to 200 μJ cm−2. Interestingly, we obtain a maximum TH-MD of 85% for EF = 300 meV and peak fluence of 200 μJ cm−2.This exceeds by far the results of ref. [5], where a similar TH-MDof 90% was obtained for a CB peak fluence of 25 mJ cm−2. Twofeatures of the data deserve to be highlighted: i) tuning EF playsa huge role in the TH-MD; ii) for all values of EF we observe anegative TH-MD.Figure 3c,d show our theoretical calculations for the THGEand TH-MD, respectively, obtained by means of the model dis-cussed in the following section. The overall agreement betweentheory and experiment is satisfactory, albeit with two shortcom-ings. The first is an overall factor in the magnitude of the signal,which can easily be traced back to an incomplete determinationof some fitting parameters, such as the attenuation of the signalin the detection apparatus, or the electron scattering rates in thetheoretical expression of the THG (see Section S6, SupportingInformation). The second is the missing ramp-up of the TH-MDat EF ≳ 250 meV. We find this discrepancy similar to what wasreported in ref. [38] in the context of the quenching of the opticalphonon-emission by Pauli blocking and attribute it to the theo-retical model missing a Fermi-energy-dependent effect that en-hances electron recombination. In any case, these two shortcom-ings do not hinder our understanding of the main feature whichwe are concerned with in the present work, namely the all-opticalswitching of the TH signal. The theoretical results fully supportour picture that the variations of the measured signal are due tothe effect of the CB on the electron distribution before the sampleis irradiated by FB.4. Theory of Ultrafast Opto-Electronic THGModulation4.1. THG Efficiency for Photoexcited ElectronsIn order to rationalize our experimental results, we need to ex-tend the theoretical treatment of the THG[8,9] to take into accountthe specific role that the CB plays in the dynamics of the elec-tron system. Indeed, the key issue of the CB-FB protocol usedin our experimental procedure is that the increase of Te, due tothe heat delivered by the CB, is inextricably linked to the pro-duction of a photoexcited electron density (𝛿ne), i.e., an excesselectron (hole) density in the conduction (valence) band. We em-phasize that such an excess carrier density is larger than the den-sity that appears in an equilibrium system when the temperatureis increased, purely due to the broadening of the Fermi-Diracdistribution across the Dirac point. Mathematically, 𝛿ne resultsin the splitting of the chemical potential (𝜇) into two differentchemical potentials 𝜇C, 𝜇V for the electrons in conduction and va-lence bands, respectively, also known as “quasi-Fermi energies”.We emphasize that the proper EF, an equilibrium quantity thatcorresponds to the value of the chemical potential at vanishingtemperature, is in a one-to-one correspondence to the electrondensity due to doping, and does not change due to process ofinter-band photoexcitation.Following refs. [8, 9], it is convenient to factor Equation (4) forTHGE asTHGE =nbn3t (nt + nb)2(IFBW0)2|S(𝜔FB + iΓe,𝜇C,𝜇V, Te)|2 (5)where nt, nb are the refractive indices of the top and bottom sub-strates, respectively, and the quantity W0 = 1012 W m−2 is intro-duced to render the expression dimensionless. Finally, the factorS is the TH conductivity, which depends on the frequency 𝜔FB ofthe FB pulse and on the thermodynamic variables of the photoex-cited electron system, i.e., Te and the two chemical potentials 𝜇Cand 𝜇V. The expression for the TH conductivity at zero temper-ature (Te = 0), in the absence of photoexcited density (𝛿ne = 0,i.e., 𝜇C = 𝜇C = 𝜖F), was given in ref. [39] in a fully analytical form,and readsS(ℏ𝜔FB + iΓe, EF) = K(EF)17G(X∕2) − 64G(X) + 45G(3X∕2)X4(6)in terms of the dimensionless function G(X) = ln [(1 + X)/(1 −X)]. The parameter K is a dimensionless constant given byK(EF) =W02𝜖20c2e4ℏv2F192𝜋E4F(7)Finally, the dimensionless quantity X= (ℏ𝜔FB + iΓe)/|EF| in Equa-tion (6) is the energy of the FB photons, rescaled by EF, and in-cludes an imaginary contribution due to the effective electronscattering rate Γe. The expression of Γe depends on the precisescattering channel responsible for the finite electron mobility,such as charged impurities, phonon, defects etc., and it mightdepend on the electron doping as well as the electron and latticetemperatures (see Section S6, Supporting Informarion).To obtain the expression of the TH conductivity of the pho-toexcited electron gas, we now apply a well-known algebraic trickdue to Maldague,[40] as detailed in ref. [41] for the linear polariza-tion function (i.e., the Lindhard function). This approach allowsus to calculate the desired quantity numerically, using an energy-integral over the analytical expression given in Equation (6):S(ℏ𝜔FB + iΓe,𝜇C,𝜇V, Te)= 14kBTe ∫∞0dE⎧⎪⎨⎪⎩S(ℏ𝜔FB + iΓe, E)cosh2(E−𝜇C2kBTe) +S(ℏ𝜔FB + iΓe, E)cosh2(E+𝜇V2kBTe)⎫⎪⎬⎪⎭−{1e−𝜇C∕kBTe + 1− 1e−𝜇V∕kBTe + 1}S(ℏ𝜔FB + iΓe, EF → 0)(8)The standard mathematical expression that relates the 𝜇V, 𝜇C, the𝛿ne, and EF can be found e.g., in ref. [8].To better illustrate the dependence of the THG on a varia-tion of electron temperature and photoexcited electron density,Adv. Sci. 2024, 11, 2401840 2401840 (5 of 9) © 2024 The Author(s). Advanced Science published by Wiley-VCH GmbH 21983844, 2024, 31, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/advs.202401840 by Kenji Watanabe - National Institute For , Wiley Online Library on [27/08/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensehttp://www.advancedsciencenews.comhttp://www.advancedscience.comwww.advancedsciencenews.com www.advancedscience.comFigure 4. TH-MD of the photoexcited electron system. The calculated value of the TH-MD as a function of Te and 𝛿ne at fixed EF equal to (a) 0.16 and(b) 0.62 in units of the FB frequency 𝜔FB. At a fixed Te, increasing the value of 𝛿ne will enhance the THG signal, while at a fixed 𝛿ne, increasing Te willreduce THG intensity. Both Te and 𝛿ne can be controlled by the incident laser power absorption.in Figure 4, we show the profile of the TH-MD (defined in Equa-tion (4)), with respect to a reference state with Te = TL and vanish-ing 𝛿ne. As expected from the equilibrium results,[39] increasingthe Te generally lowers the value of the PTH (i.e., negative TH-MD). Increasing the 𝛿ne, on the contrary, increases the PTH, ascan also be expected from the doping-dependence known fromthe equilibrium results.[39] In other words, 𝛿ne can be seen as aquasi-equilibrium electron- and hole-doping in conduction andvalence band, respectively. It follows that the CB can affect theTHG in two competing ways because it produces a Te increasethat is necessarily coupled to the production of 𝛿ne. It is thennecessary to know the precise relation between Te(t) and 𝛿ne(t)in time to predict the THG following a given CB. To this end, weresort to the solution of a model dynamics, based on a simplerate-equation approach, which we outline in the following sec-tion.Before we discuss our dynamical model, we remark that theprocedure that leads to Equation (8) cannot be applied to arbi-trary non-equilibrium states of the electron system, but assumesthat the carriers in the two bands are thermalized to the same Te,although it allows for two different 𝜇V and 𝜇C. Mathematically,this means that the electron (hole) distribution in the conduction(valence) band is given by a Fermi-Dirac function of the formfe,h(E,𝜇C,V(t), Te(t)) =1e(E±𝜇C,V(t))∕kBTe(t) + 1(9)where the carrier energy E is measured from the Dirac point.Thequasi-equilibrium assumption of Equation (9) then holds if thesystem’s dynamics is coarse-grained on a time-scale longer thanthe electron thermalization time-scale, which has been shown tobe shorter than ≈20 fs in graphene.[35] The dynamical model thatwe adopt here is fully consistent with this limitation.4.2. Model Dynamics of Photoexcited ElectronsTo model the dynamics of photoexcited electrons, we adopt a rate-equation approach that describes: i) electron heating due to thelaser beams; ii) energy exchange between electrons and opticalphonons, due to emission and absorption processes; iii) opti-cal phonon relaxation to the lattice equilibrium temperature (seee.g., ref. [38] and references therein). The variables of interest arethe Te(t), 𝛿ne(t), and the occupation of the optical phonon modesaround the center of the Brillouin zone (Γ point) and the valleys(K points), with frequencies 𝜔Γ and 𝜔K, respectively.The time-derivative of the Te is given by the net absorbed powerdivided by the heat capacitydTe(t)dt=(t) − RΓ(t)ℏ𝜔Γ − RK (t)ℏ𝜔Kce(t) + ch(t)(10)where (t) = FB(t) + CB(t) is the average power absorbed perunit area, ce, h(t) are the electron and hole heat capacity per unitarea, and RΓ, K(t) are the net phonon emission and absorptionrates. The expressions for the electron absorbance (that relatesthe absorbed to the incident power in the linear regime) andthe heat capacity can be found e.g., in ref. [8]. Here, we calcu-late the heat capacity as the sum of the electron and hole con-tribution, taken into account independently, because inter-bandrecombination processes are much slower than thermalization,and thus do not contribute to the temperature adjustment, whichis mathematically described by the heat capacity coefficient. Thephonon rates follow from a standard Boltzmann formula that canbe found e.g., in ref. [38]. We remark that the coefficients dis-cussed above depend on the electron distribution and phonon oc-cupation, and must thus be calculated dynamically in time as thesystem evolves. Notwithstanding its simple appearance, Equa-tion (10) is a strongly non-linear equation of motion.The time derivative of the 𝛿ne is given by the number of pho-tons absorbed minus the number of phonons emitted by inter-band transitions, per unit time and aread𝛿ne(t)dt=CB(t)ℏ𝜔CB+FB(t)ℏ𝜔FB− RΓ,inter(t) − RK,inter(t) (11)Adv. Sci. 2024, 11, 2401840 2401840 (6 of 9) © 2024 The Author(s). Advanced Science published by Wiley-VCH GmbH 21983844, 2024, 31, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/advs.202401840 by Kenji Watanabe - National Institute For , Wiley Online Library on [27/08/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensehttp://www.advancedsciencenews.comhttp://www.advancedscience.comwww.advancedsciencenews.com www.advancedscience.comNotice that photon absorption always results in an interband tran-sition. We remark that the 𝛿ne depends on interband phononemission rate only, while the Te depends an all phonon emis-sions: this is obviously because all phonon emissions reduce en-ergy but only interband phonon emissions reduce the 𝛿ne.Finally, the rate equations for the phonon occupation are eas-ily obtained by requiring consistency with Equations (10) and (11)in terms of energy and particle balance. Typical results of the in-tegration of these rate equations are reported in the Section S5(Supporting Information).5. DiscussionWhen a laser pulse is incident on a graphene flake, Te increasesover the pulse duration (see Section S5, Supporting Information)until it reaches a steady-state condition. In ref. [8] we safely useda steady-state condition in order to attribute the changes in THsignal to a single value of Te for a fixed value of EF. This holds aslong as one pulse measurement is performed on the graphene.In order to dedicate a single value to Te, either an instantaneousvalue or a value after the relaxation of the electrons (ps range)must be considered. However, considering the pulse durationsused in our study of 110 to 150 fs, limits us from both consid-erations. So this intermediate state in terms of pulse durationenables us to estimate a minimum and maximum Te for the ex-perimental values in Figure 3. At EF = 50 meV and CB fluenceof 11 and 200 μJ cm−2, we estimate a Te in the range ≈1500 to1900 K and ≈2300 to 2500 K, respectively. At a higher value ofdoping (EF = 300 meV), we estimate a Te in the range ≈800 to1300 K for the CB fluence of 11 μJ cm−2 and Te ≈ 2200 to 2300 Kfor the CB fluence of 200 μJ cm−2.Furthermore, the origin of the TH enhancement reported inFigure 1 resulting from a reduction in TL can be attributedto two coherent and incoherent physical processes. First, spec-tral broadening induced by FB leads to band broadening andalters carrier lifetimes, thereby affecting the THGE. Second,the well-established thermodynamics of carriers involving relax-ation of carriers through optical phonons, which is temperature-dependent, contribute to the change in THGE. In other words,the significant impact of TL on the TH modulation can be qual-itatively understood based on two mechanisms, which includethe dependence of electronic spectral broadening Γe and kineticrelaxation rates R(t) on lattice temperature TL. The temperaturedependence of Γe predominantly originates from the scatteringof electrons by acoustic phonons, while kinetic rates depend ontemperature due to the electron-optical phonon interaction.Finally, it is worth highlighting the interplay between the Teand photoexcited enhanced Pauli blocking. Steady-state theoreti-cal considerations in ref. [8] predict that at low values of doping(when EF < ℏ𝜔/2), increasing Te will lead to the enhancement ofthe THG signal, a result that we were never able to observe exper-imentally in this work. However, these steady-state predictionsrely on the assumption that 𝛿ne remains constant once grapheneis irradiated with a pulsed laser. In contrast, Figure 4 shows howthe evolution of the TH-MD is accompanied by both the Te and𝛿ne changes, both quantities that play a key role in the presence ofboth FB and CB, as discussed above. Thus, for instance, Figure 4ashows the evolution of TH-MD when EF/(ℏ𝜔FB) is 0.16. For lowervalues of doping (corresponding to Te ≈ 1500 to 2500 K in ourexperiments) and (𝛿ne < 1012cm2), TH-MD is always negative.This indicates that 𝛿ne is not large enough to compete with thehigh Te, which is consistent with the experimental observationsin Figure 3. On the other hand, when EF/(ℏ𝜔FB) is 0.62, nega-tive TH-MD occurs for Te > 1300 K (Figure 4b). Considering theTe that we reach during the experiments (1500 to 1900 K) at thisregime of doping, TH-MD is still negative. This also confirmsthat 𝛿ne in our experiments never reaches more than 1012cm2,where TH-MD would turn positive. It is worth mentioning thatby comparing Figure 4a,b, one can immediately notice that thechange in TH-MD as a function of Te is smaller when EF/(ℏ𝜔FB)is 0.62. This behavior is consistent with the results in ref. [9].Therefore THG in graphene is always accompanied by the twocompeting and interconnected effects of Te (hot electrons) and𝛿ne (Pauli blocking).To conclude, it is worth mentioning that our results couldbe readily applied to other gapless systems such as bilayergraphene[42] and surface states of topological insulators.[43] Fur-thermore, this work provides an interesting benchmark for thetuning of the optical nonlinearities in quantum confined sys-tems. For instance, thermal, electrical and all-optical tuning ofharmonic generation has been widely studied in transition metaldichalcogenides (TMDs), where, however, the modulation mech-anism has a completely different physical origin compared tographene. In the case of thermal modulation, Khan et al.[44] re-ported an enhancement of the second harmonic (SH) intensityin a MoSe2 monolayer of ≈25% when tuning the lattice tem-perature from ≈153 to 393 K and for a non-resonant excitationwavelength of 900 nm. This SH modulation was attributed toa thermal expansion of the lattice (i.e., changes in the distancebetween atoms), and thus completely different with respect tothe thermal modulation of graphene TH reported in this work,which we attribute to changes in the electronic distribution. Alsoin the case of electrical modulation, the physical mechanisms atplay are completely different between graphene and TMDs. Ingraphene, the external gate voltage enables tuning of the Fermienergy across multi-photon resonances in the Dirac cone.[8,21] Incontrast, electrical modulation in TMDs is mainly due to tun-ing of the optical resonances from neutral to charged excitons(trions).[6] Furthermore, gate tuning also leads to different re-combination dynamics both in graphene and TMDs. In TMDs,such changes are attributed to the different lifetime of trions withrespect to neutral excitons.[45] In contrast, in graphene the gatetuneable recombination dynamics are due to quenching of thescattering between hot electrons and optical phonons.[38] For theall-optical modulation of harmonic generation in TMDs, we haverecently demonstrated a modulation scheme for SHG[4] whichfully exploits the crystal symmetry and it allows to rotate the po-larization of the emitted SH signal by 90° on ultrafast timescales.This ultrafast SH tuning can be exploited, in combination withspecifically designed metasurfaces, to achieve light wavefrontshaping.[46] A second scheme for all-optical modulation exploitsPauli blocking. This is possible both in graphene, as demon-strated in this work, and in TMDs.[7] Here, the main differenceis that all-optical modulation by Pauli blocking can be realized inprinciple at any wavelength in graphene, due to its linear bandabsorption and broadband absorption, while in TMDs all-opticalmodulation of the NLO response is efficient only at resonancewith excitonic transitions.[47]Adv. Sci. 2024, 11, 2401840 2401840 (7 of 9) © 2024 The Author(s). Advanced Science published by Wiley-VCH GmbH 21983844, 2024, 31, Downloaded from https://onlinelibrary.wiley.com/doi/10.1002/advs.202401840 by Kenji Watanabe - National Institute For , Wiley Online Library on [27/08/2024]. See the Terms and Conditions (https://onlinelibrary.wiley.com/terms-and-conditions) on Wiley Online Library for rules of use; OA articles are governed by the applicable Creative Commons Licensehttp://www.advancedsciencenews.comhttp://www.advancedscience.comwww.advancedsciencenews.com www.advancedscience.com6. ConclusionIn conclusion, we performed a detailed experimental and theo-retical study of static thermal and ultrafast opto-electronic mod-ulation of THG in a high-quality graphene FET encapsulated inthin hBN layers. We have demonstrated static switching of THGvia tuning of the lattice temperature and electron doping, witha peculiar ambipolar behavior that arises from the electron-holesymmetry in the Dirac cone, and a factor of ≈1.5 modulation ofthe PTH at EF = 50 meV and of ≈3 at EF = 300 meV when tuningthe TL from room temperature to 33 K. We suggest that this resultoriginates from the spectral relaxation and thermodynamic kinet-ics of carriers. Furthermore, we have established all-optical ultra-fast control of graphene THG with gate tunable dynamics, andachieved up to 85% ultrafast opto-electronic modulation depth ofthe TH at EF = 300 meV and fluence of 200 μJ cm−2, which is twoorders of magnitude more efficient compared to previous reports.We discuss the EF dependent temporal dynamics of all-opticalTH modulation due to quenching of the phase-space scatteringbetween optical phonons and electrons.[30] This provides a power-ful tool to actively control both the TH modulation depth and therecombination dynamics in graphene opto-electronic nonlineardevices. Finally, we have addressed these experimental observa-tions with a detailed theoretical framework that explains the ul-trafast opto-electronic modulation of TH in graphene to be rootedin a mixed effect of Pauli blocking and carrier electronic tempera-ture. Consequently, our work can be seen as the first step towarda holistic approach for manipulating not only THG, but morein general any third-order non-linear optical effect in graphene,such as saturable absorption for synchronized dual-fiber lasers[48]or THz pulse generation.[49] Furthermore, our findings can beused to optimize the performances of nonlinear optical devicesfor applications in gas sensing[11] and logic gates,[12] and for thedevelopment of hybrid photonic devices for the enhancement ofoptical nonlinearities, for instance in fiber-[50] and waveguide-based frequency converters.[51] Thus, thanks to a detailed descrip-tion of the transient nonlinear optical, electronic, and thermal re-sponse of graphene, this work will be relevant for the design of awide range of nanoscale and ultrafast nonlinear optical devices.Supporting InformationSupporting Information is available from the Wiley Online Library or fromthe author.AcknowledgementsA.T. acknowledged the “National Centre for HPC, Big Data and QuantumComputing”, under the National Recovery and Resilience Plan (NRRP),Mission 4 Component 2 Investment 1.4 funded from the European Union- NextGenerationEU. H.R. acknowledged the support from the SwedishResearch Council (VR Starting Grant no. 2018-04252). K.W. and T.T. ac-knowledged support from the JSPS KAKENHI (Grant Numbers 21H05233and 23H02052) and World Premier International Research Center Initia-tive (WPI), MEXT, Japan. G.S. acknowledged the German Research Foun-dation DFG (CRC 1375 NOA), project number 398816777 (subproject C4)and the International Research Training Group (IRTG) 2675 “Meta-Active”,project number 437527638 (subproject A4).Open access funding enabled and organized by Projekt DEAL.Conflict of InterestThe authors declare no conflict of interest.Data Availability StatementThe data that support the findings of this study are available from the cor-responding author upon reasonable request.Keywordsall-optical THG modulation, electrically tunable THG, graphene, nonlinearopticsReceived: February 22, 2024Revised: May 8, 2024Published online: June 18, 2024[1] O. Dogadov, C. Trovatello, B. Yao, G. Soavi, G. 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Introduction 2. Ambipolar Gate-Tunable THG 3. Ultrafast Opto-Electronic TH Modulation 4. Theory of Ultrafast Opto-Electronic THG Modulation 4.1. THG Efficiency for Photoexcited Electrons 4.2. Model Dynamics of Photoexcited Electrons 5. Discussion 6. Conclusion Supporting Information Acknowledgements Conflict of Interest Data Availability Statement Keywords