Progress Report www.advopticalmat.de Terahertz Nonlinear Optics of Graphene: From Saturable Absorption to High-Harmonics Generation Hassan A. Hafez, Sergey Kovalev, Klaas-Jan Tielrooij, Mischa Bonn, Michael Gensch, and Dmitry Turchinovich* visible light frequencies, corresponding Graphene has long been predicted to show exceptional nonlinear optical to interband transitions leading to elec- properties, especially in the technologically important terahertz (THz) tron–hole pair generation.[5,6,12] In the frequency range. Recent experiments have shown that this atomically thin case of doped graphene, that is, gra- material indeed exhibits possibly the largest nonlinear coefficients of any phene containing a certain concentra- tion of free electrons or holes, Drude-like material known to date, paving the way for practical graphene-based applica- conductivity was observed at far-infrared tions in ultrafast (opto-)electronics operating at THz rates. Here the advances and terahertz (THz) frequencies, corre- in the booming field of nonlinear THz optics of graphene are reported, and sponding to intraband free-carrier absorp- the state-of-the-art understanding of the nature of the nonlinear interaction of tion.[8,9,12–15] Further, theoretical studies graphene with the THz fields based on the thermodynamic model of electron predicted strong nonlinear interaction of graphene with intense light fields, in transport in graphene is described. A comparison between different mecha- particular at the technologically impor- nisms of nonlinear interaction of graphene with light fields in THz, infrared, tant THz frequencies (typically, in the and visible frequency ranges is also provided. Finally, the perspectives for the range 0.1–10 THz),[16–25] as originating expected technological applications of graphene based on its extraordinary from both interband and intraband elec- THz nonlinear properties are summarized. This report covers the evolution tron dynamics. These predictions were of the field of THz nonlinear optics of graphene from the very pioneering to inspired by the unique band structure of graphene: absence of a bandgap and the state-of-the-art works. It also serves as a concise overview of the cur- linear energy–momentum dispersion for rent understanding of THz nonlinear optics of graphene and as a compact its electrons.[2,3,26–28] A plethora of strong reference for researchers entering the field, as well as for the technology nonlinear effects in graphene in the IR developers. and optical frequency ranges, originating from interband electron dynamics, was successfully demonstrated, including saturable absorption and nonlinear refraction,[29–39] higher- 1. Introduction harmonic generation,[40–47] and wave-mixing processes[48–50] (see also reviews[51–53]). At THz frequencies, however, until The advent of single-layer graphene[1–3] and ultrathin epi- recently only saturable absorption effects in doped gra- taxial graphite[4] in 2004 stimulated extensive research of phene,[54–61] and induced multiphoton-like absorption in the optical properties of graphene over a broad range of the multilayer near-intrinsic graphene were successfully dem- electromagnetic spectrum.[5–12] These studies revealed uni- onstrated.[62] At the same time, the observation of the long versal absorption (or optical conductivity) at infrared (IR) and sought-after effect of THz higher-order harmonics generation, Dr. H. A. Hafez, Prof. D. Turchinovich Dr. S. Kovalev Fakultät für Physik Helmholtz-Zentrum Dresden-Rossendorf Universität Bielefeld Bautzner Landstraße 400, 01328 Dresden, Germany Universitätsstr. 25, 33615 Bielefeld, Germany Dr. K.-J. Tielrooij E-mail: dmtu@physik.uni-bielefeld.de Catalan Institute of Nanoscience and Nanotechnology (ICN2) Dr. H. A. Hafez, Prof. M. Bonn BIST & CSIC Max-Planck-Institut für Polymerforschung Campus UAB, Bellaterra 08193, Barcelona, Spain Ackermannweg 10, 55128 Mainz, Germany Prof. M. Gensch The ORCID identification number(s) for the author(s) of this article Institut für Optische Sensorsysteme can be found under https://doi.org/10.1002/adom.201900771. DLR Rutherfordstraße 2, 12489 Berlin, Germany © 2019 The Authors. Published by WILEY-VCH Verlag GmbH & Co. KGaA, Prof. M. Gensch Weinheim. This is an open access article under the terms of the ­Creative Institut für Optik und Atomare Physik Commons Attribution License, which permits use, distribution and Technische Universität Berlin ­reproduction in any medium, provided the original work is properly cited. Strasse des 17. Juni 135, 10623 Berlin, Germany DOI: 10.1002/adom.201900771 Adv. Optical Mater. 2020, 8, 1900771 1900771 (1 of 25) © 2019 The Authors. Published by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.advancedsciencenews.com www.advopticalmat.de that is, the multi­ plication of the frequency of incident THz photons in graphene, remained elusive. This situation has Hassan A. Hafez obtained his changed recently when very efficient generation of THz M.Sc. degree from Helwan harmonics up to the seventh order in single-layer-doped University, Egypt, in 2008, graphene was demonstrated using multicycle quasi-mono- and his Ph.D. degree from chromatic THz driving fields, revealing gigantic nonlinear the University of Quebec optical coefficients of graphene in the THz range.[63] The (INRS-EMT), Canada, in 2016. physical mechanism of the demonstrated highly efficient Then, he became a lecturer THz high-harmonics generation (THz HHG) and THz and faculty staff member at saturable absorption in doped graphene was attributed to Helwan University, followed by the collective thermodynamic response of the background a postdoctoral stay at MPI for Dirac electrons of the doped graphene to the THz driving Polymer Research in Germany, field, which is currently believed to be the dominating effect in and a position as research interaction of free electrons in graphene with THz fields. The associate at the University of Duisburg-Essen. In 2019, underlying physical picture of this interaction, and its effect he was appointed akademischer Rat at Bielefeld University on the THz conductivity and absorbance in graphene will be where he is currently leading the development of high-inten- provided within this report. Finally, we emphasize the impor- sity terahertz facility. His research interests include nonlinear tance of exploring the THz nonlinear properties of graphene terahertz optics and ultrafast dynamics in condensed matter. for emerging technologies. Graphene-based transistors,[64–67] photodetectors,[68,69] saturable absorbers for laser mode- locking,[30,70,71] and other high-speed (opto-)electronic devices, Michael Gensch obtained his all usually operate in the nonlinear regime, and are entering Ph.D. degree from the TU the THz frequency/rate operation mode. Berlin in 2004. After post- doctoral stays at the Leibniz- Institute of Analytical Sciences 1.1. How Is This Report Organized? and Deutsches Elektronen- Synchrotron and a posi- Our report is organized in five sections. After the introduction tion as staff scientist at the in section 1, section 2 provides a brief overview of the impor- Helmholtz-Zentrum Berlin, he tant fundamentals of graphene, such as its band structure and joined the Helmholtz-Zentrum key electronic properties. Section 3 deals with the linear inter- Dresden-Rossendorf as action of graphene with light fields in general, accounting for project leader for the TELBE the interband and intraband electron dynamics. Section 4 pre- Terahertz facility in 2010. In 2015, he became group leader sents (i) the experimental observations of nonlinear THz optics for “High-Field Terahertz-Driven Phenomena”. In 2019, of graphene and other graphene-based systems, and a brief he became professor of physics at the TU Berlin and head overview of the experimental techniques and facilities for the of the “Terahertz and Laser Spectroscopy” department at generation of intense THz fields, (ii) the thermodynamic model the DLR Institute of Optical Sensor Systems. His research that describes the nonlinear interaction of graphene with interests include terahertz spectroscopy, ultrafast science, intense THz fields, and (iii) the experimental generation of instrumentation development, and 2D materials. THz high harmonics, which directly follows from the thermo- dynamic picture of THz electron transport in graphene. Where Dmitry Turchinovich relevant, the results on graphene under optical excitation will graduated from St. Petersburg also be briefly discussed. Finally, section 5 provides conclusions Electrotechnical University and future perspectives on possible applications of graphene as and Ioffe Institute in 1999 and a highly nonlinear THz material. obtained his Ph.D. from the University of Freiburg in 2004. After a postdoctoral stay at 2. Band Structure of Graphene Utrecht University, in 2006 he joined the faculty at Technical Graphene is a monolayer of carbon atoms bound covalently in University of Denmark. In a hexagonal honeycomb lattice, as shown in Figure 1a. Each 2012–2017 Dmitry Turchinovich carbon atom in the graphene plane is bound to three nearest led the Ultrafast Dynamics & neighbor atoms via strong equivalent σ-bonds established by Terahertz Spectroscopy group at MPI for Polymer Research sp2 hybridization of three of the valence electron-orbitals of each in Mainz. In 2017 he was appointed professor of physics at atom, namely, 2s, 2px, and 2py,[26–28,72] as shown in Figure 1b. the University of Duisburg-Essen, and in 2018 he became These in-plane σ-bonds are responsible for the mechanical full professor of physics at Bielefeld University. His research strength of graphene. The fourth electron-orbital 2pz is oriented interests are ultrafast and terahertz dynamics in condensed perpendicular to the graphene plane. The overlap between the matter, and general ultrafast science. out-of-plane 2pz orbitals of the carbon atoms results in weaker Adv. Optical Mater. 2020, 8, 1900771 1900771 (2 of 25) © 2019 The Authors. Published by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.advancedsciencenews.com www.advopticalmat.de Figure 1. Graphene lattice and band structure: a) Hexagonal honeycomb lattice of graphene, b) binding between the carbon atoms in the graphene lattice, c) the triangular sublattices A and B, d) the first hexagonal Brillouin zone exhibiting two inequivalent groups of Dirac points, and e) the energy- momentum spectrum of the graphene band structure described by Equation (1). Note the rotation of k-axes with respect to (d). (Reproduced, except for (b), with permission.[26] Copyright 2009, American Physical Society). π-bonds. These delocalized π-electrons are responsible for the is indicative of the positive kinetic energy of electrons in the electrical conduction in graphene. conduction band (the π* electrons) and the negative (binding) The hexagonal lattice of graphene is considered as a bipartite energy of the bound electrons (the π electrons) in the valence lattice composed of two triangular sublattices, marked A and B band. This unique linear dispersion ε = vF ℏk with constant in Figure 1c. The sublattices are the Bravais lattices of graphene band velocity in the vicinity of the Dirac points in graphene is with lattice vectors a1 = 0.5a( 3, 1) and a2 = 0.5a( 3, − 1) and a characteristic of ultrarelativistic massless quasiparticles: Dirac lattice constant a = |a1| = |a2| = 0.246 nm.[27] We note here that fermions with zero rest mass. The linear band structure approx- the lattice constant a is related to the carbon–carbon distance imation of Equation (2) is valid for the energies up to about acc = 0.142 nm by a = 3 acc . ± 1.5 eV with respect to the Dirac point, thus covering a very In the reciprocal space, the first Brillouin zone is con- broad range of energies (frequencies) of the electromagnetic sequently a hexagon featuring six corner points of two spectrum from THz to UV. inequivalent groups of K-points called the Dirac points indic- The density of states in graphene D(ε) within the linear part ative of the two sublattices A and B, as shown in Figure 1d. of the band structure is a linear function of energy, with D = 0 The corresponding band structure is shown in Figure 1e, in at the Dirac point ε = 0 which the valence band touches the conduction band at these 2 six Dirac points, providing a gapless band structure of gra- D (ε ) = ε (3) phene. This band structure of graphene was first calculated π  2 v F2 by Wallace in 1947 using the tight-binding approximation,[73] The Fermi energy EF depends on the free carrier density Nc providing cosine-like energy bands in energy–momentum in graphene as space[27] E F = v F π N c (4) 1/2   3kx a   ky a   ky a   Very importantly for the electronic transport in graphene, ε (kx , ky ) = ± γ 0 1 + 4cos  cos  + 4cos2  (1)   2    2    2   it can be shown that for the “massless” electrons with linear energy–momentum dispersion such as Equation (2) it is fun- where γ0 ≈ 2.8 eV is the hopping energy (transfer integral) damentally not possible to electrostatically bind them to a between the nearest-neighbor atoms. Near the Dirac points particle of opposite charge. This is in stark contrast to the “mas- where |k|a ≪ 1, the Taylor series expansion of the energy sive” charge carriers with quasi-parabolic energy–momentum spectrum of Equation (1) reduces to the linear dispersion ­dispersion, such as electrons and holes in regular metals or relation[27] semiconductors, which can be trapped by charged impuri- ties, or form excitons. This absence of electrostatic (Coulomb) 3 ε± (k ) = ± γ 0 ak = ± v F k (2) binding for its electrons also makes graphene stand out among 2 other 2D materials with “massive” quasi-parabolic energy where v F = 3 γ 0 a /2 ≈ 10 6 m s−1 is the energy- and momentum- momentum dispersion for the charge carriers, such as transi- independent band velocity (Fermi velocity)[73] and the ± sign tion metal dichalcogenides, which feature extremely strong Adv. Optical Mater. 2020, 8, 1900771 1900771 (3 of 25) © 2019 The Authors. Published by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.advancedsciencenews.com www.advopticalmat.de electronic Coulomb binding due to the very weak dielectric corresponding to such intraband transitions, in case of a ther- (lattice) screening, leading in some cases to exciton binding mally equilibrated electron population, is described by the fol- energies as high as ≈500 meV.[74] lowing solution of the Boltzmann transport equation[28,59] In graphene, due to the fundamental absence of Coulomb ∞ τ (ε ) ∂ f FD (ε , µ ,Te ) e 2 v F2 binding, electron transport is not affected by the trapping onto σ intra (ω ) = − 2 ∫ D (ε ) 1 − iωτ (ε ) ∂ε dε (6) impurities and other unscreened Coulomb potentials, and is 0 only limited by the momentum scattering. This unique pro­ where τ(ε) is the energy-dependent electron momentum scat- perty of graphene electrons, combined with the large optical tering time, and fFD(ε, μ, Te) is Fermi–Dirac distribution func- phonon energy of graphene of the order of ≈200 meV, and tion depending on the electron energy ε, chemical potential µ, the nonpolar nature of its lattice, leads to a very large carrier and electron temperature Te. mobility that can exceed 100 000 cm2 V−1 s−1 observable even at For the case of Te ≪ EF/kB, which corresponds to a sharp room temperature.[66,72,75] Fermi edge, only the electron scattering time at the Fermi Further, graphene exhibits a strong ambipolar field effect energy, τ = τ(EF), matters, and Equation (6) reduces to the fol- when subjected to a gating voltage, allowing for a precise con- lowing Drude-type expression[8,9,82] trol of the type and density of the free charge carriers (electrons 2e 2τ or holes) in the respective bands.[1] Combined with naturally σ intra (ω ) = kBTe ln [2 cosh(E F /2kBTe )] (7) high electron mobility, this made graphene a promising can- π  (1 − iωτ ) 2 didate for the development of ultrafast transistors operating at The THz excitation of graphene is dominated by the intra- the rates of 100s of GHz.[64,65] band conductivity (Equations (6) and (7)), especially at room temperature. Indeed, the photon energy in the few meV range (1 THz = 4.1 meV) is considerably smaller than the room- 3. Light–Matter Interaction in Graphene in the temperature thermal energy kBTe = 26 meV. As a result, in intrinsic, undoped graphene with EF ≈ 0, at room temperature, Linear Regime the thermal population of the conduction and valence bands Depending on the photon energy ℏω and the Fermi energy becomes almost equal: (fFD, conduction − fFD,valence) |1THz, 300K = 0.04 of graphene EF, the excitation of graphene with light will lead for the interband transition at the frequency of 1 THz = to either interband or intraband transitions. The pioneering 4.1 meV. This leads to effective Pauli-blocking of the interband observation and distinction between both types of transitions in THz transitions in graphene at room temperature, making graphene was made by Winnerl et al.[9] Optical excitation of gra- them only possible under cryogenic conditions (see, e.g., phene with ℏω > 2EF results in an interband transition, similar ref. [62]). However, in practical THz experiments on large area, to photoexcitation of semiconductors, and leads to the creation ≈mm sized, single-layer graphene samples, even at lower tem- of an electron–hole pair, as shown schematically in Figure 2a. peratures the THz interband transitions are not easy to reveal The interband transitions in graphene exhibit a universal, that because of the presence of so called “charge puddles,” local is, frequency-independent, absorption A = πα = 2.3% of the areas of n- or p-doping with |EF|> 0, again Pauli-blocking the incident light,[5] where α = (1/4πε0) e2/ℏc = 1/137 is the fine- interband THz transitions.[83] structure constant, corresponding to the universal optical con- For the typical doping levels for substrate-supported gra- ductivity σ0 = e2/4ℏ  1 G. Such a universal light absorption phene, resulting in a Fermi energy on the order of 100 meV is a direct consequence of the unique electronic properties (see, e.g., refs. [59,63,72,84–90]), the interband THz transitions of graphene: the linear energy–momentum dispersion and are Pauli-blocked even at T = 0, leaving only the intraband tran- the gapless band structure. We note that for photon energies sitions described by the free carrier conductivity of Equation (6) below ≈0.5 eV and above ≈2 eV, a deviation from this universal possible. absorption occurs.[6,12,76] The frequency dependence of the cor- We note that most single-layer graphene samples are in responding interband conductivity was shown to follow the fact unintentionally doped due to impurities induced in the relation[6,8–10,12] preparation process as well as the effects of the environment in which graphene is used.[72,85–90] For example, in the course e2   ω + 2E F   ω − 2E F   σ inter (ω ) = tanh  + tanh  (5) of preparation of chemical vapor deposition (CVD) grown 8    4kBTe   4kBTe   graphene, which is one of the most popular graphene forms where kB is the Boltzmann constant and Te is the electron tempera- broadly available nowadays, hole-doping is induced during the ture. We note that for large photon energies, that is, ℏω ≫ 2EF and ℏω transfer of poly(methyl methacrylate) (PMMA)-covered gra- ≫ kBTe, Equation (5) reduces to the universal conductivity σ0 = e2/4ℏ. phene from copper foils to dielectric substrates by residues of We further note that the optical absorption (and reflection) of PMMA.[90] The dielectric substrate itself, and water and oxygen graphene depends strongly on its dielectric environment.[77,78] molecules trapped between the graphene and the substrate For photons of energy ℏω < 2EF, as is typically the case ­further contribute to the hole-doping of CVD graphene.[90,91] On at THz frequencies, the interband transitions are Pauli- the other hand, in epitaxially grown graphene on SiC electron- blocked,[9,12,79–81] whereas the intraband transitions are possible, doping is induced by the SiC substrate (substrate-induced cha as depicted schematically in Figure 2b. Such transitions rging).[9,27,86,92,93] For multilayer epitaxial graphene, the layers correspond to the interaction of the incident light with the elec- closer to the middle of the sample can be intrinsic with EF ≈ 0, trons around the Fermi surface of graphene, that is, to the free while outer layers are usually doped by the environment and carrier absorption. The frequency-dependent optical conductivity the substrate.[9,62,93] Adv. Optical Mater. 2020, 8, 1900771 1900771 (4 of 25) © 2019 The Authors. Published by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.advancedsciencenews.com www.advopticalmat.de Figure 2. Interaction of graphene with a light field: Schematic of the band structure with a) interband transition due to absorption of a photon of energy larger than twice the Fermi energy, ℏω > 2EF, leading to electron–hole pair generation, followed by the energy and momentum exchange with the Fermi sea electrons via electron–electron scattering, and b) intraband dynamics when ℏω < 2EF, which exhibits Drude-like conductance. c) Demonstration of the transmission of a THz field through a graphene sample with absorbed portion of the field ΔETHz related to the graphene intraband conductivity (Reproduced under the terms of the Creative Commons CC-BY 4.0 License.[59] Copyright 2015, Springer Nature). d,e) Actual measurement of THz fields and the corresponding spectra, respectively, transmitted through a graphene sample on SiO2 substrate and through a bare SiO2 substrate as a refer- ence, and f) the corresponding complex THz intraband conductivity normalized to the universal conductivity σ0 = e2 /4ℏ, extracted from the complex spectra (the Fourier transforms) of the measured fields in (d) using Equation (8) and described by the Drude-like intraband conductivity of Equation (7) using EF = 170 meV and τ = 47 fs as fitting parameters (Reproduced with permission.[63] Copyright 2018, Springer Nature). Given the symmetry of Dirac band structure of graphene spectra of graphene. With THz time-domain spectroscopy (see Figure 1), there is no fundamental difference between (THz-TDS),[94,95] which is based on direct time-domain sampling the electron and hole doping: Both charge species can be con- of the propagating THz electric field, one can extract the gra- sidered equivalent and conduct electric currents in the same phene conductivity from the experimentally measured THz field fashion. In the literature, for simplicity, the Fermi level is often transmission of the graphene film through the ­following rela- indicated within the conduction band even if graphene is in tion in the frequency domain (known as Tinkham equation)[96] fact hole-doped (such as typical CVD-grown graphene). Corre- spondingly, irrespective of the actual type of majority carriers E s (ω ) ns + 1 T (ω ) = = (8) in graphene, they are often referred to as electrons. In this  E ref (ω ) ns + 1 + Z0σ (ω ) paper, we follow this convention, unless specifically indicated. Figure 2c–f shows the THz field transmission through a where ns is the refractive index of the substrate at the THz fre- monolayer CVD-grown p-doped graphene sample, and the quencies, Z0 = 377 Ω is the free-space impedance, and E s (ω ) corresponding THz power transmission and conductivity and E ref (ω ) are the spectral complex Fourier transforms of the Adv. Optical Mater. 2020, 8, 1900771 1900771 (5 of 25) © 2019 The Authors. Published by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.advancedsciencenews.com www.advopticalmat.de Table 1. Simplified overview of the fundamental timescales of optical fields and electron dynamics in nonintrinsic (EF > 0.1 eV) graphene. Spectral range and ­corresponding Light field oscillation Electron transport lifetime Electron–electron Electron–SCOPa) Electron–phonon typical light-wave frequency and period (average electron momentum ­scattering scattering scattering wavelength scattering time) (thermalization time) NIR: 200 THz, 1.5 µm 5 fs 10–100 fs 10–100 fs 10–100 fs ≈1 ps THz: 1 THz, 300 µm 1 ps a)SCOP = strongly coupled optical phonons. measured time-domain fields Es(t) and Eref(t), respectively, as by Mikhailov,[16,17,19] who initially used semiclassical calculations shown in Figure 2d. Here, Es(t) is the THz field transmitted and ballistic transport. Later theories were extended to include through graphene on a substrate, and Eref(t) is that trans- quantum mechanical considerations,[18,21,22] and more recent the- mitted through a bare substrate of identical thickness used as ories of graphene nonlinearities furthermore take into account a reference. Figure 2f shows the complex conductivity, real and realistic conditions regarding momentum scattering time and imaginary, obtained from the experimental THz fields shown ambient temperature. Interestingly, the effect of the elevated elec- in Figure 2d at room temperature, and fitted by the Drude-like tron temperature on graphene nonlinearities was not considered conductivity of Equation (7). The only adjustable parameters in of significant importance in these earlier works. In 2015, Mics this fit are the Fermi energy EF and the momentum scattering et al.[59] showed that the strong THz nonlinearity in graphene is time τ. Thus, THz-TDS allows one to unambiguously deter- the result of THz-induced carrier heating, which was followed by mine the key parameters of electron transport in graphene in several other studies, culminating in the observation of extremely the linear regime—the number density of conducting charge efficient THz high-harmonics generation in graphene in 2018 Nc, linked to EF by Equation (4), and the average momentum by Hafez et al.[63] Regarding the visible and near-infrared (NIR) charge carrier scattering time τ. Note that the sheet conductivity regime, the role of the increased carrier temperature on graphene (i.e., conductivity times thickness) in units of S = Ω−1 is used in nonlinearity was first addressed in 2018 by Soavi et al.[46] and Equation (8). Baudisch et al.[97] These studies show that properly taking into account the electron temperature leads to a significant adjust- ment in the theoretically predicted nonlinearities, yielding excel- lent agreement between theory and experiment. 4. Terahertz Nonlinear Interactions in Graphene The nonlinearity in graphene is naturally related to the trans- 4.1. General Considerations on THz and Optical/IR port of its electrons in the applied electric field of an incident ­Nonlinearities in Graphene light wave. In order to understand the fundamental differ- ence between graphene nonlinearities in the THz regime and Since the advent of graphene, several theories were put forward in the optical or NIR regime, we first summarize the relevant that predict a highly nonlinear response of this material, in timescales of incident light wave oscillations and of electron particular in the THz regime. This field of research was initiated dynamics in graphene (see Table 1 and Figure 3). Figure 3. Illustration of nonlinear graphene–light interaction at NIR and THz frequencies: a) Illustration of the nonlinear graphene electron current response (red) from incident near-infrared (NIR) light at 200 THz (blue), after Baudisch et al.[97] The dominant dynamics are based on the coherent photon–electron interactions. Since the induced electron current does not exactly follow the incident electric field, redshifted harmonics are generated. b) Illustration of the nonlinear graphene heat response (red) from incident THz light at 1 THz (blue), after Hafez et al.[63] The dominant dynamics are based on the heating–cooling dynamics of the graphene–electron system, which modulate the graphene intraband conductivity and consequently the induced electron current, leading to re-emission at THz higher harmonics. The lengths of the black lines under the figure panels indicate the funda- mental timescales of the relevant processes, scaling with the corresponding figure above them. Adv. Optical Mater. 2020, 8, 1900771 1900771 (6 of 25) © 2019 The Authors. Published by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.advancedsciencenews.com www.advopticalmat.de We remind that since Coulomb trapping for massless Dirac In contrast to IR or optical light waves, THz waves oscillate fermions is fundamentally forbidden, electronic transport on a (sub-)picosecond timescale, considerably longer than the in graphene is dominated by momentum scattering events. typical electron momentum scattering time in graphene which Graphene electrons undergo momentum scattering by long- is on the order of 10–100 fs. Therefore, in the THz range the range and short-range interactions with charged impurities, carrier transport typically occurs in the diffusive regime, where disorder, and phonons.[28] In substrate-supported graphene, the electrons undergo several momentum scattering events within former typically dominates, and the momentum scattering time the duration of the THz wave oscillation period (see Table 1), is typically on the order of 10s fs, corresponding to mobilities in and the microscopic motion of electrons is generally inco- the range of 1000–10 000 cm2 V−1 s−1. If the energy is supplied herent with the driving electric field. As we will explain in to the electron system, this energy is ultimately redistributed detail, the nonlinearity of graphene related to such diffusive among all the carriers by electron–electron scattering, leading collective motion of free electrons in the applied THz fields is to an increase of (collective) electron temperature. Whereas this in fact extremely strong, leading to substantial THz saturable process depends on several parameters, it typically takes some absorption[9,55,57,59] and extremely efficient THz high-harmonics 10s fs.[98–100] Highly excited electrons with the energy exceeding generation.[63] Remarkably, this nonlinearity mechanism only that of the optical phonon energy of ≈0.2 eV can also relax by requires THz fields of the order of 10s of kV cm−1, which is interaction with strongly coupled optical phonons (SCOPs), ≈3 orders of magnitude smaller (i.e., requiring about one million which is also believed to occur on the timescale of some 10s times less peak power) than that required for the optical or IR fs.[100] For a moderately excited electron system (electron temper- nonlinearity in graphene via coherent Bloch oscillations.[47,108] ature of ≈1000 K), carrier relaxation (i.e., cooling) occurs via cou- In general, THz nonlinearities in graphene can originate from pling to (i) graphene acoustic phonons through disorder-assisted both intraband and interband THz transitions. Simple consid- scattering,[101] (ii) graphene–optical phonon interactions,[102] and erations, however, point out that the THz intraband nonlinear (iii) substrate phonons through near-field interactions.[103,104] At response is expected to be much stronger than the interband one. room temperature, these cooling processes typically take place First, the number of states available for the interband THz tran- on a timescale of picoseconds, although longer timescales are sitions is fairly small. Integration of the density of states in gra- predicted to occur in the absence of disorder and a substrate.[105] phene (Equation (3)) within 1 THz = 4.1 meV band around the The nature of the nonlinear response in graphene Dirac point yields only about N ≈ 108 cm−2 electrons available for can be divided into ballistic and diffusive, depending on interband THz transitions. On the other hand, a typical doping the timescale of electron transport lifetime (i.e., average of graphene readily yields a Fermi energy EF ≈ 100 meV and free momentum scattering time) and the duration of the optical carrier density of Nc ≈ 1012 cm−2 (see Equation (4)),[59,63,85] avail- cycles of the driving light wave. Very strong nonlinearity in able for the interaction with the THz fields via intraband con- the electronic motion in graphene in the ballistic regime ductivity, and leading to a typical THz power absorbance on the can be achieved making use of the coherent Bloch oscil- order of Aintra ≈ 0.1 at room temperature (see, e.g., refs. [59,63]). lations.[106,107] They arise from the acceleration of an elec- Second, as explained in Section 3, the Pauli-blocking of interband tron from the Brillouin zone center to the Brillouin zone THz transitions at elevated temperatures strongly suppresses the edge, with ∆k = π , in the electric field of the light wave, in a interband absorption of graphene, leading to a negligible interac- a time faster than the average electron momentum scattering tion with the THz fields. The room-temperature power absorb- time τ = 10–100 fs, thus making electrons oscillate coher- ance of graphene at 1 THz frequency is only Ainter = πα(fFD, con- ently between Δk and −Δk points in the Brillouin zone (here −3 duction − fFD, valence)|1THz, 300K ≈ 10 , which is ≈100 times smaller a = 0.246 nm is the lattice constant of graphene). In the optical than the typical intraband THz absorbance for doped graphene of or IR ranges, the duration of the optical cycle is on the order Aintra ≈ 0.1. Here, α is the fine-structure constant (see Section 3). of a few femtoseconds, which is faster than the average elec- Therefore, THz nonlinearities related to interband transitions tron momentum relaxation time in graphene (see Table 1), thus in graphene can only be observed in (almost perfectly) intrinsic ensuring the ballistic motion of an electron on the timescale graphene, and only at cryogenic temperatures. Indeed, nonlinear of the light wave oscillation. From the acceleration theorem multi-photon-like induced THz absorption in near-intrinsic gra- dk ∆k phene was recently demonstrated under such conditions, origi-  = − eE , which can be further simplified to  = − eE , it follows dt τ nating from a combination of coherent and incoherent inter- and that a minimum electric field on the order of 1–10 MV cm−1 is intraband excitation mechanisms.[62] However, this experiment required to reach the regime of coherent Bloch oscillations in required a multilayer sample (45 layers) in order to significantly graphene with a typical scattering time of τ = 10–100 fs. Here, enhance the light–matter interaction strength. ℏk is crystal momentum, t is time, e is elementary charge, and At the same time, the intraband transitions in graphene E is electric field strength of the light wave. Such rather high remain active even at room temperature and, as mentioned peak fields exceeding 1–10 MV cm−1 are routinely attainable for above, lead to strong THz absorbance with Aintra ≈ 0.1. As a optical and IR signals generated with amplified laser systems. result, the interaction of graphene with THz waves via the Indeed, high-harmonics generation in graphene using coherent intraband conductivity mechanism remains much stronger in Bloch oscillations mechanism was recently demonstrated with both linear and nonlinear regimes, as compared to the inter- IR fields of 10s MV cm−1 strength by Yoshikawa et al.[47] and band THz transitions.[9,13,55,59,63,111] Higuchi et al.[108] We note that coherent Bloch oscillations can Figure 4 shows the state-of-the-art of the experimentally be induced at lower field strength in systems with reduced measured third-order nonlinear susceptibility χ(3) of graphene, Brillouin zone, such as semiconductor superlattices.[109,110] in the spectral range from visible to THz. A significant increase Adv. Optical Mater. 2020, 8, 1900771 1900771 (7 of 25) © 2019 The Authors. Published by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.advancedsciencenews.com www.advopticalmat.de experiments represent the nonlinear (variable field strength) version of THz time-domain spectroscopy (THz-TDS), THz pump–THz probe, or THz pump–optical probe experiments, and are performed with spectrally broadband, near single-cycle THz pulses. A comprehensive description of high-field table- top THz sources and nonlinear THz spectroscopic technics can be found in refs. [61,122]. Further, an important role in understanding of the IR and THz optical properties of graphene was played by THz sources at large-scale accelerator-based facilities, such as IR/THz free electron lasers (FELs)[123] and superradiant THz undulator facil- ities,[124] providing spectrally dense, narrowband, and broadly tunable output. The most common experiment at FELs has been based on degenerate one-color pump–probe spectroscopy. In a series of careful experimental campaigns, the induced tran- sient differential transmission change in graphene has been measured on Fourier-limited few picosecond timescales over a wide range of frequencies stretching from few THz to mid- infrared frequencies.[9,114,125,126] These experiments revealed important insights into the carrier dynamics in graphene, Figure 4. Experimentally demonstrated third-order susceptibilities from including the pioneering observation of inter- and intraband the NIR to the THz range. Susceptibilities from the following references transitions in doped graphene.[9] are plotted over the wavelength of the exciting light pulses: [a] Dremetsika and Kockaert,[112] [b] Kumar et al.,[43] [c] Soavi et al.,[46] [d] Hendry et al.,[48] Most recently, a prototype superradiant THz undulator [e] Kundys et al.,[113] [f ] König-Otto et al. (Landau-quantized graphene in facility TELBE[124] was used for the pioneering demonstration a magnetic field),[114] and [g] Hafez et al.[63] Whereas in the NIR (left), of THz HHG in single-layer graphene.[63] The key for the suc- the nonlinearity mechanism is typically associated with coherent elec- cess of this experiment was a unique combination of emission tron dynamics, in the THz (right) the mechanism is noncoherent and and detection characteristics of TELBE: (i) spectrally pure, thermal. For intermediate wavelengths, different mechanisms have been multicycle, phase-stable strong-field pulses of THz and sub-THz proposed, such as plasmon nonlocal effects [e], and thermally induced plasmon shifts[115,116] (no susceptibility quoted). radiation delivered at exceptionally high repetition rates, and (ii) novel THz detection schemes based on precise femtosecond of χ(3) with the fundamental wavelength of the driving light timing of the superradiant undulator facility to external laser field is observed, yielding the largest nonlinear coefficient of sources. The former allowed for exceptionally good ­statistics χ(3) ≈ 10−9 m2 V−2 at the THz frequencies (case [g] in Figure 4). on the measurement, and the latter enabled a particularly This value of χ(3) of graphene is several orders of magnitude larger sensitive probing of nonlinear THz signals in the time-domain than that of other materials in any spectral range, possibly making with subcycle temporal resolution and a dynamic range as graphene the most nonlinear material known to date. We note that high as 106. even when normalized to material thickness (in case of graphene d = 0.3 nm)[114] and using the sheet values for nonlinear coeffi- cients χ(3)2D = χ(3) d, the nonlinearity of graphene still by far exceeds 4.2.2. Key Observations of Nonlinear THz Absorption in Graphene that of other known highly nonlinear systems: χ 2D (3) ≈ 10 −19 m3 V−2 (3) 3 −2 for graphene versus χ 2D ≈ 10 m V for highly nonlinear −23 Here, we will present the key observations of nonlinear THz quantum well structures with d of a few nm.[117,118] absorption in graphene, as well as several results of optical pump–THz probe spectroscopy that have been crucial for the 4.2. Experimental Observation of THz Nonlinearities understanding of the nature of THz nonlinearity of graphene. in Graphene In 2011, Hwang et al.[54,55] reported the pioneering observa- tion of the THz nonlinearity of CVD-grown graphene, using In the following, we will highlight the exemplary experimental intense single-cycle THz pulses with a maximum fluence of results and measurement approaches that led to the state-of- 190 µJ cm−2 corresponding to a peak electric field strength of the-art understanding of the THz nonlinearity in graphene. ≈100 kV cm−1, see Figure 5a–d. As experimental methods in this study, nonlinear THz-TDS and THz pump–THz probe spectroscopy were used, as depicted schematically in Figure 5a. 4.2.1. Experimental Techniques In this work, a THz-field-induced transparency of doped gra- phene, that is, an increase of THz transmission through gra- Various techniques and different sources of intense THz fields phene with increasing the THz pump field, shown in Figure 5b,c, have been employed for nonlinear THz spectroscopy of gra- was demonstrated for the first time. The nonlinear transmission phene. The most commonly used strong-field THz sources enhancement was attributed to saturable absorption effects are table-top sources based on optical rectification of ampli- related to a decrease in the intraband conductivity of the doped fied laser pulses in various nonlinear crystals,[55,57–59,62,119,120] or graphene sample. The measured power-dependent THz trans- based on air-plasma THz generation method.[121] Most of such mission of graphene was described, as shown in Figure 5c, by Adv. Optical Mater. 2020, 8, 1900771 1900771 (8 of 25) © 2019 The Authors. Published by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.advancedsciencenews.com www.advopticalmat.de Figure 5. THz field-induced transparency in monolayer graphene: a) Schematic of varying high-field THz-TDS and intense-THz-pump/weak-THz- probe spectroscopy of graphene on a substrate; techniques followed by Hwang et al.[55] b) Spectral field transmission increasing with the THz fluence, with the temporal THz fields (left inset) and the corresponding spectra (right inset) transmitted through the graphene sample and the bare substrate, c) The power transmission versus the THz pump fluence; symbols for the experiment and solid red line for the saturable absorption fit, d) Transient differential transmission of the THz probe beam transmitted through THz-pumped graphene as a function of the pump–probe delay time, showing ultrafast (less than 100 fs) transient increase in transmission followed by a subsequent biexponential relaxation over a picosecond time range (b–d: Adapted with permission.[55] Copyright 2013, American Chemical Society), and e) suppression of graphene conductivity with THz field for all the THz frequencies in test demonstrating the observed transparency (Reproduced under the terms of the Creative Commons CC-BY 4.0 License.[59] Copyright 2015, Springer Nature). a phenomenological saturable transmission function yielding variation of the electron momentum scattering time as a free a saturation fluence of 20 µJ cm−2. An attempt to reproduce fitting parameter: The trend of reduction of THz conductivity the data quasi-quantitatively using a Drude model of the THz of graphene could be reproduced by reducing the electron scat- intraband conductivity of graphene (see Equation (7)) required tering time in graphene with increasing the driving THz field. Adv. Optical Mater. 2020, 8, 1900771 1900771 (9 of 25) © 2019 The Authors. Published by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.advancedsciencenews.com www.advopticalmat.de In the THz pump–THz probe experiment presented in the ℏω = 30 meV. Increasing the pump fluence, as expected, results same work,[54,55] the dynamics of THz absorption bleaching in in a larger differential transmission ΔT/T signal. graphene was demonstrated, with quasi-instantaneous increase We note here the k-space distribution of the photoexcited in THz transmission followed by a biexponential relaxation carriers in graphene is initially anisotropic. This anisotropy on ≈ 200 fs and ≈picosecond timescales (see Figure 5d). The was revealed in pump–probe FEL experiments by exam- ­pioneering observation of THz nonlinearity of doped graphene ining the polarization relation between the pump and probe by Hwang et al.[54,55] was followed by further experiments on beams,[93,126,127] as shown in Figure 6d,e. It is also observable nonlinear THz spectroscopy of graphene[56,58,59,62,119,120] using for a broad range of IR excitation photon energies and even at table-top intense near single-cycle THz pulse sources, and THz frequencies,[128] for as long as a sufficient pump fluence is similar observations of THz-field-induced transparency (THz available, for both inter- or intraband excitations. This anisot- absorption bleaching) in both CVD[56,58,59,111] and epitaxial[57,119] ropy vanishes via carrier relaxation by phonon emission within graphene were reported. The physical mechanism of THz sub-picoseconds or on a longer timescale, depending on the absorption bleaching was mainly attributed in all these works energy of the excited state relative to the optical phonon ener- to nonlinear intraband conductivity modulation (suppression) gies in graphene.[93,126] Excitations in lightly doped graphene by the intense THz driving field (see Figure 5e). However, the below the optical phonon energy (≈200 meV) are accompanied precise mechanism of THz conductivity suppression remained by a longer preservation of such an anisotropy over picoseconds under debate at the time. due to quenching of the optical phonon emission.[126,127] In 2014, Bowlan et al.[62] observed induced absorption and In contrast to the case of high photon energy inducing nonlinear spectral modification of near single-cycle broadband interband transitions, pumping at lower photon energies (i.e., strong-field THz pulses in near-intrinsic epitaxial 45-layer gra- ℏω < 2EF) in one-color pump–probe FEL experiment leads to phene sample at a cryogenic temperature, which was assigned interband absorption of the probe beam: negative probe differ- to a complex process initiated by nonlinear THz interband ential transmission ΔT/T0 for ℏω = 20 meV in Figure 6a. This transitions in graphene, vanishing with increasing the sample is a result of smearing out the carrier distribution in the band temperature. In a more recent work, four-wave mixing in by heating the carriers via free-carrier absorption (intraband multilayer (50 layers) epitaxial graphene in strong magnetic dynamics) of the pump beam, thus opening channels for the field, and driven by intense far-IR field at 19 THz was reported probing photons of the same energy to induce interband transi- by König-Otto et al.[114] tions, as illustrated in Figure 6c. We note that this latter obser- In 2015, Mics et al.[59] published a combined experimental- vation of stronger THz probe absorption in excited graphene theoretical study presenting a thermodynamic picture of the seems to be in qualitative disagreement with the observations of THz nonlinearity in graphene. This work largely reconciled THz probe transmission enhancement by Hwang et al.[55] shown the previous observations of THz nonlinearities in graphene, in Figure 5d. This discrepancy can nevertheless be accounted and provided a quantitative description of the mechanism of for by the much lower doping concentration of the 70-layer the strong modulation of intraband conductivity in the THz- graphene sample, and the relatively large photon energies excited graphene using common conservation laws. The basic ℏω = 20 meV = 4.9 THz employed in the study by Winnerl et al. mechanism of THz nonlinearity of graphene, which will be Indeed, according to the explanation in Section 3, for such high described in detail in Section 4.3, is as follows. The absorption photon energies the Pauli-blocking of interband transitions at of the THz field by free electrons of graphene leads to quasi- room temperature becomes relaxed, thus permitting measurable instantaneous increase in the electron temperature, accompa- interband THz absorption in a low-doped multilayer sample. nied by a downshift in the chemical potential, which in turn Let us now consider the optical pump–THz probe experi- suppresses the intraband conductivity of graphene.[58,62,81] ments, which produced a bulk of valuable knowledge, and This rather simple thermodynamic model of the THz nonlin- significantly contributed to the present understanding of the earity of graphene by Mics et al.[59] is currently believed to be nature of THz nonlinearity of graphene. In the following, we the most accurate one, showing very good agreement with the will notice the similarities between the THz response of gra- experimental results on picosecond-timescale THz nonlinear phene to strong THz excitation and to optical excitation. optics of graphene. Optical excitation of graphene significantly modifies the With one-color pump–probe spectroscopy using a tun- electronic occupation of its band structure. Absorption of an able FEL, in 2011 Winnerl et al.[9,126] studied ultrafast carrier optical pulse by graphene leads to the creation of initial elec- dynamics in quasi-intrinsic (lightly doped) multilayer tron–hole pairs, which thermalize among all charge carriers (70 layers) epitaxial graphene grown on SiC at a range of photon within ≈100 fs, leading to a transient electron distribution with energies from 10 meV (≈2.5 THz) to 250 meV (≈62.5 THz), and an elevated temperature, that is, to carrier heating. This ultra- made the pioneering observation of, and distinction between, fast electron heating in photoexcited doped graphene, that is, the inter- and intraband transitions in graphene. For photon the smearing out of electron Fermi–Dirac distribution in the energies larger than twice the Fermi energy (i.e., ℏω > 2EF), energy–momentum space, was first directly revealed in 2013 1 in a pioneering time-resolved angular-resolved photoemission pump-induced occupation of energy states that are ω above 2 spectroscopy (tr-ARPES) experiment by Gierz et al.[100] per- the Dirac point via interband transition leads to an absorp- formed with both inter- and intraband IR excitations. tion bleaching of the probe beam similar to semiconductors, An important excursion must be made here. If the electronic resulting in a positive differential transmission signal (increase temperature becomes high enough (≈3000 K), this photo­ in transmission), as shown in Figure 6a–c for the case of induced carrier heating also leads to a weak (due to impedance Adv. Optical Mater. 2020, 8, 1900771 1900771 (10 of 25) © 2019 The Authors. Published by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.advancedsciencenews.com www.advopticalmat.de Figure 6. One-color pump–probe spectroscopy of epitaxial multilayer graphene using narrow-band tunable free-electron laser pulses: a) Differential transmission of the probe beam as a function of the pump–probe delay time at various pump fluences for two cases of photon energy; ℏω = 30 meV > 2EF and ℏω = 20 meV < 2EF (Reproduced with permission.[126] Copyright 2017, Wiley-VCH, while the original content of the figure is published in Ref. [9]), b) model calculations of the corresponding real conductivity at different electron temperatures over a range of photon energies, showing pump-induced absorption of the probe beam when ℏω < 2EF and induced transparency when ℏω > 2EF (Reproduced with permission.[126] Copyright 2017, Wiley-VCH), c) schematic explanation of the revealed responses in (a) and (b) (Reproduced with permission.[9] Copyright 2011, American Physical Society), d) schematic of anisotropic distribution of photoexcited carriers accumulated in the direction perpendicular to the polarization direction of the pump beam (Adapted with permission.[127] Copyright 2014, American Chemical Society), e) differential transmission signal revealing the anisotropy described in (d) using one-color pump–probe spectroscopy in which parallel and perpendicular polarizations of the pump and probe beams were compared (Reproduced with permission.[93] Copyright 2016, American Physical Society). mismatch with the far-field), very short-living (<100 fs), yet However, if the photoexcited graphene is significantly doped, observable, spectrally broad optical emission, essentially repre- the situation becomes quite different. For a typical doping senting the blackbody radiation by the photogenerated hot elec- level with EF > 100 meV, several groups[14,15,84,119,120,129,131,132] tron population in graphene. This effect was discovered in 2010 observed a reduction of THz probe absorption of doped gra- by Lui et al.[130] in pristine graphene under 800 nm optical exci- phene as a result of photoexcitation. In Figure 7, we show the tation. In particular, the high-frequency tail of the blackbody exemplary results by Tielrooij et al.[129] and Jensen et al.[84] emission, with photon energies exceeding the excitation photon This negative photoconductivity is in a way counterintuitive, energy of 1.5 eV, was observed with sensitive photodetectors in as it shows that ultrafast optical interband excitation in fact the range 1.7–3.5 eV.[130] This optical emission due to the black- makes doped graphene more THz-resistive. In a combined body radiation is a collective process of the hot carrier electron experimental-theoretical study in 2013, Tielrooij et al.[129] distribution, and is thus very different from the conventional assigned this phenomenon to “hot-carrier multiplication,” a semiconductor-type radiative electron–hole recombination. For cascaded electron–electron scattering process, triggered by the example, this blackbody optical emission, unlike electron–hole energetic photoexcited electrons, which scatter with the back- recombination in semiconductors, does not lead to a reduction ground electrons in the Fermi sea and increase their energy. in the density of excited carriers in graphene. The presence of The microscopic theory for such hot-carrier multiplication such blackbody emission in photoexcited graphene,[130] and by multiple electron scattering in graphene was presented by the previously discussed time-resolved ARPES measurements Song and Levitov in the work of Song et al.,[98] where it was also showing the ultrafast increase in electronic temperature in demonstrated that the rate of such electron–electron scattering photoexcited graphene,[100] together provided convincing proof generally largely exceeds the electron–phonon scattering rate. of the occurrence of photoinduced carrier heating in graphene. Such a photoinduced hot-carrier multiplication was connected For near-intrinsic or lightly doped graphene with EF of the order in the work of Tielrooij et al.[129] (see its Supporting Informa- of a few tens of meV,[84,119,120] ultrafast photoexcitation leads to an tion) to very fast electron thermalization and hence to the increase of the THz-probed graphene conductivity as a result of increase of the electronic temperature in graphene, leading to the increase in free carrier density, and observed as negative dif- the reduction of its chemical potential and conductivity. These ferential transmission of the probe THz pulse.[131,132] This effect studies, explaining the negative photoconductivity of doped gra- is similar to above-bandgap photoexcitation of semiconductors. phene, became quite important also for the understanding of Adv. Optical Mater. 2020, 8, 1900771 1900771 (11 of 25) © 2019 The Authors. Published by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.advancedsciencenews.com www.advopticalmat.de Figure 7. Optical pump–THz probe spectroscopy. a) Schematic of the experiment (Reproduced with permission.[129] Copyright 2013, Springer Nature), b) THz differential transmission of the peak of the THz probe field as a function of the pump–probe delay time for two cases of optical-pump photon energies maintaining the same photoexcited carrier density of ≈1011 cm−2, showing an increase in the THz transmission of graphene after photoexci- tation, corresponding to a suppression of the photoconductivity (Adapted with permission.[129] Copyright 2013, Springer Nature). c) Crossover from negative to positive photoconductivity in graphene with decreasing free carrier concentration. Negative photoconductivity in doped graphene and positive photoconductivity in near-intrinsic graphene at fixed photoexcitation level. A sample with a back-gate was used in order to control the position of the Fermi level in graphene (Adapted with permission.[84] Copyright 2014, American Chemical Society). THz-induced nonlinearities in graphene. In Section 4.3, we will (static doping) and ultrafast optical excitation, the situation show that the final result of both intense THz (with an electric became more complex.[84,119,120,131,132] Depending on the ini- field strength exceeding 10 kV cm−1, fluence of ≈2 µJ cm−2) and tial doping density and the THz probe field strength, the THz optical excitation (usually with a fluence exceeding 1 µJ cm−2) photoconductivity was found to be either negative (THz absorp- of doped graphene is, in fact, equivalent—an increase of the tion bleaching) or positive (enhancement of THz absorption), temperature of its background electron population, which in as shown in Figure 8d–f. In lightly doped graphene with EF on turn reduces its intraband, that is, THz conductivity.[59,63,84,129] the order of a few meV, increased THz absorption in photo­ The doping concentration in graphene was found to play a excited graphene was noticed when probing by weak THz significant role in defining the THz response of graphene to field (see Figure 8e), similar to the results in ref. [84] shown in both optical and intense THz excitations. Razavipour et al.[111] Figure 7c, indicating the semiconductor-like THz response of reported the effects of the carrier concentration in graphene on near-intrinsic graphene. With increase in the THz probe field the THz field-induced nonlinearity (only absorption bleaching), strength, THz absorption bleaching with a crossover to nega- using gated graphene samples in which the doping concentra- tive THz photoconductivity (positive THz differential trans- tion was controlled via electrochemical gating, as depicted sche- mission) was observed. On the other hand, for strongly doped matically in Figure 8a. It was found that the THz field-induced graphene the THz photoconductivity always remained negative, transparency in graphene (or the THz nonlinearity in general) indicative of THz absorption bleaching (positive THz differen- becomes more pronounced with increasing the doping concen- tial transmission) as a result of combined action of THz and tration, as shown in Figure 8b,c. This indicates that the non- optical excitation, as shown in Figure 8f. linear THz field effects in graphene are dominated by intra- In a more recent work, Tomadin et al.[133] significantly band conductivity mechanisms, so that the higher the doping extended the understanding of the effect of photoexcitation on concentration the larger the possible modulation of the intra- graphene based on a combination of microscopic modeling and band conductivity by the THz field is. optical pump–THz probe spectroscopy. Interestingly, it was found When the free carrier concentration in the sample was that depending on the Fermi energy of the sample, the ultrafast controlled by the combination of electrical doping by gating electronic thermalization in photoexcited graphene can lead Adv. Optical Mater. 2020, 8, 1900771 1900771 (12 of 25) © 2019 The Authors. Published by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.advancedsciencenews.com www.advopticalmat.de Figure 8. Carrier density dependence of the THz field-induced nonlinearity in graphene: a) Schematic of the interaction of intense THz field with Fermi-level-tunable graphene, b,c) nonlinear THz transmission of gated graphene as a function of both the THz field strength and the doping level expressed by the gating voltage relative to the voltage corresponding to the charge neutrality point V–VCNP (Reproduced with permission.[111] Copyright 2015, American Physical Society). d) Schematic of optical pump-induced dynamics probed by intense THz field, and e,f) peak of THz differential transmission signal ΔT/T0|peak of photoexcited graphene as a function of the THz probe field for lightly doped graphene pumped by 29 µJ cm−2 optical pump fluence (Adapted with permission.[120] Copyright 2015, AIP Publishing), highly doped graphene at three optical pump fluences of 29, 79, and 137 µJ cm−2 (Adapted with permission.[119] Copyright 2015, American Physical Society). not only to “hot-carrier multiplication,” that is, electron heating the interband optical absorption of the 1.5 eV probe pulse in with the total number of free carriers conserved (case of larger graphene. The impact ionization in strong (up to 300 kV cm−1), EF > 0.1 eV), but under certain conditions also to a “real” free ultrafast (≈30 fs) THz field pulses as observed by Tani et al.[121] carrier multiplication via interaction with the valence band states increases the free carrier density in graphene. This is an addi- (case of smaller EF < 0.1 eV). Carrier multiplication in graphene tional effect to the THz-induced electron heating—the main was first predicted by Winzer et al.[134] and observed experimen- effect of interaction of doped graphene with intense THz fields, tally by Plötzing et al.[135] Ref. [133] furthermore provides insights which as such conserves the free carrier density (see Section 4.3 into the mechanism leading to the reduction of the graphene for details). To the best of our knowledge, the impact ionization conductivity with increasing electron temperature, based purely leading to carrier multiplication has not been observed in the on electron kinetics effects, including the effect of screening, and experiments dealing with “slower” (>1 ps long) strong-field THz considering that momentum scattering is dominated by long- pulses as used in the experiments of refs. [56–59,63,111,119]. range Coulomb scattering with impurities outside graphene. Besides standard “continuous” graphene samples, non- An interesting observation was made by Tani et al.[121] in a linear THz spectroscopy was also performed on patterned THz pump–optical probe experiment, performed using ≈30 fs graphene, featuring the confinement of THz field, and thus single-cycle THz pump pulse and NIR probe of 800 nm wave- leading to enhancement of the THz nonlinearity. For example, length (1.55 eV photon energy). A standard doped CVD-grown it was theoretically predicted[136–138] and experimentally veri- graphene was used in this work. It was shown that the excita- fied[51,115,116] that enhancement of the nonlinear interaction of tion of graphene with the THz pulses with electric field of the light field in graphene can be achieved by local field con- up to 300 kV cm−1 resulted in an ultrafast (≈150 fs) transient finement via resonant plasmonic absorption if the graphene increase of the optical probe transmission through graphene, is patterned in subwavelength structures such as ribbons. as shown in Figure 9. This effect was assigned to absorption Using THz FEL, Jadidi et al.[115,116] demonstrated nonlinear bleaching of the optical probe by Pauli-blocking of the NIR THz absorption at plasmonic resonance in lithographi- interband transitions at 1.55 eV photon energy under intense cally patterned epitaxial graphene ribbons (see Figure 10a), ultrafast THz excitation of graphene. The authors attributed the following one-color pump–probe scheme with THz field electron occupation of states around 0.775 eV above the Dirac polarization perpendicular to the patterning direction. The point to impact-ionization effects. The impact ionization is the plasmonic resonance frequency (see Figure 10b) was defined process of electron–hole pair generation resulting from the loss by the dimensions of the graphene pattern, as well as by the of kinetic energy of another highly energetic electron or hole, electron density and temperature. The THz pump at reso- which was assumed to be ballistically accelerated by the strong nance resulted in two consequent effects: (i) a rise in the pump THz field. It was suggested that this process leads to car- electron temperature by the THz excitation enhanced due to rier multiplication, allowing for the occupation of higher energy the THz field confinement, and consequently (ii) a redshift in states around 0.775 eV above the Dirac point, in turn bleaching the resonance absorption. The probe experienced absorption Adv. Optical Mater. 2020, 8, 1900771 1900771 (13 of 25) © 2019 The Authors. Published by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.advancedsciencenews.com www.advopticalmat.de Figure 9. THz pump/optical probe of single layer graphene: a) Schematic demonstration of the THz field induced dynamics in hole-doped CVD gra- phene, showing ballistic acceleration imposing impact ionization from the valence band to the conduction band leading to occupation of states that block further interband transitions by the optical probe beam, b) the THz pump field with a peak field of 300 kV cm−1 with the corresponding spectrum shown in the inset, c) negative differential optical density corresponding to enhanced transmission of the optical probe beam as a function of the pump–probe delay time, d,e) experimental data (symbols) of the optical density change due to the intense THz pump as a function of the pump–probe delay time, and model calculations (solid lines) based on the Boltzmann equation with scattering mechanisms including impact ionization and Auger recombination for THz field strengths of 170 and 300 kV cm−1, respectively, f,g) calculated carrier density following the impact ionization and Auger recombination, as a function of the pump–probe delay time, and h) the peak field dependence of the peak of the differential optical density, showing 2.5 and a better fit with the model only when impact ionization and Auger recombination were considered (Reproduced with permis- a scaling with E THz sion.[121] Copyright 2012, American Physical Society). Figure 10. Plasmonic-induced enhancement of THz nonlinearity in lithographically patterned graphene ribbons: a) Schematic of lithographically patterned graphene ribbons on SiC substrate with a ribbon periodicity L and a ribbon width W. b) Broadband THz transmission through the sample showing plasmonic absorption resonance at 3.8 THz (the dark cyan line with arrow indicating the relevant axis). The normalized spectrum in red corresponds to one of the FEL pulses (here at resonance) employed in the degenerate pump–probe spectroscopy (with arrow indicating relevant axis) (Replotted with data provided by the authors and with permission.[116] Copyright 2019, American Chemical Society). c) THz differential transmis- sion at resonance of the probe beam as a function of the pump–probe delay time for various THz pump fluences (Reproduced with permission.[116] Copyright 2019, American Chemical Society), d) The peak differential transmission signal as a function of the pump fluence; colored symbols for the experiment and a solid black line for the F fit (Reproduced with permission.[116] Copyright 2019, American Chemical Society), e) the differential transmission signal at various frequencies around the resonance frequency showing a sign reversal when the probe frequency is tuned to be below resonance (Reproduced with permission.[116] Copyright 2019, American Chemical Society), and f) the peak of the differential signal in (e) as a function of frequency (Reproduced with permission.[116] Copyright 2019, American Chemical Society). Adv. Optical Mater. 2020, 8, 1900771 1900771 (14 of 25) © 2019 The Authors. Published by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.advancedsciencenews.com www.advopticalmat.de bleaching, with positive differential transmission signal ΔT/T0 reduction of the THz conductivity of graphene. The effect of THz increasing as the square root of the pump fluence, as shown excitation of doped graphene is thus the same as the effect of in Figure 10c,d. The relatively small pump fluence employed optical excitation (also summarized in Section 4.2.2), which simi- here, as compared to that employed in the studies with con- larly reduces the THz, that is, intraband, conductivity of graphene. tinuous planar graphene,[9,55,59] showed a significant enhance- In the following, we will describe the basic mechanism of ment in the THz field-induced nonlinearity by resonant plas- such a THz-driven conductivity suppression, following the ther- mons in patterned graphene. On the other hand, the redshift modynamic model of ultrafast charge transport in graphene of the plasmon resonance due to the rise in the electron tem- presented by Mics et al. in 2015.[59] This model, resting on basic perature induced by the THz pump led to increased absorp- conservation laws, self-consistently treats the THz-induced tion of the probe beam when probing at frequencies lower electron heating in doped graphene during its interaction with than the original resonance frequency (negative ΔT/T0 in the incident THz field, dynamically modifying the propagation Figure 10e,f). conditions for the THz field itself. The key physical processes described by the thermodynamic model of THz nonlinearity of graphene are summarized in Figure 11. 4.3. The Thermodynamic Model of Nonlinear Intraband THz We note here that most equations of this model are only Response of Graphene strictly valid in the Sommerfeld expansion regime, that is, at elec- tron temperature being below the Fermi temperature: Te < TF, The observations of the nonlinear THz response of graphene, where TF = EF/kB . For highly doped graphene on SiO2 with described in Section 4.2.2, can be summarized as follows. For typical Fermi energy of 0.25–0.4 eV, the Fermi temperature is doped graphene, excitation with intense THz fields (field strength TF = 3000–4800 K. For practical purposes, however, the approxi- above ≈10 kV cm−1, fluence of the order of ≈µJ cm−2) leads to a mation Te < TF in the description of nonlinear interaction of Figure 11. The thermodynamic model of THz-driven electron transport in graphene: a) Schematic for the THz field-induced electron dynamics via intra- band THz conductivity (THz absorption by the Fermi surface electrons) showing intraband heating enabled by e–e scattering that allows electrons in the band to thermalize among themselves, which is accompanied by a reduction of the chemical potential to keep the carrier density constant (Reproduced under the terms of the Creative Commons CC-BY 4.0 License.[59] Copyright 2015, Springer Nature), b) the population density before (blue) and after (red) the THz excitation (Reproduced with permission.[63] Copyright 2018, Springer Nature), c) the corresponding Fermi–Dirac distribution before and after the THz excitation showing a smearing-out of the hot carrier distribution in the band accompanied by a downshift in the chemical potential to maintain the total carrier density (the area under the curves in (b)) constant, and d) illustration of spectral weight conservation for inter- and intraband transitions in graphene. Reduction of chemical potential for hotter electron population leads to increase of the spectral weight for interband transitions (as a result of Pauli-unblocking), which is compensated by the reduction of the spectral weight of intraband conductivity in graphene (c,d: Reproduced with permission.[82] Copyright 2015, IOP Publishing). Adv. Optical Mater. 2020, 8, 1900771 1900771 (15 of 25) © 2019 The Authors. Published by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.advancedsciencenews.com www.advopticalmat.de arbitrary THz field with graphene remains reasonably accurate. Since the electrons are assumed as internally thermalized at Most of the nonlinear response in graphene is in fact achieved all times, and hence are described at all times by Fermi–Dirac in the regime of Te < TF, and the effect of further increase in   ε − µ  distribution f FD (ε , µ , Te ) = 1/ exp  kBTe  + 1 with a certain tempera- electron temperature appears smaller in comparison. For ture Te and chemical potential µ, the free electron conductivity higher electron temperatures, leading to chemical potential of a THz-driven graphene at all times can be described by approaching the Dirac point, the accuracy of the thermody- Equation (6) depending on the above parameters. Quantita- namic model by Mics et al.[59] can be improved by including the tively, the total thermal energy density (energy per unit area) valence band states, as shown by Frenzel et al.[132] of the graphene electronic system excited by THz field can be Let us first consider the energy conservation in the interaction expressed as Q 0 + ∆Q = ∫ 0 εD(ε ) f FD (ε , µ ,Te )dε , where D (ε ) = 2 ε /π ( v F )2 ∞ of THz radiation with doped graphene. The application of the is the density of states. Here, Q0 is the electronic thermal THz field ETHz to graphene leads to generation of THz currents energy before the THz field excitation, that is, with the chemical according to Ohm’s law, j = σintra ETHz, where σintra is the intra- potential μ equals to the initial Fermi energy at the initial tem- band conductivity of graphene, described by Equation (6). Gener- perature (e.g., room temperature). With the THz field absorp- ally, the conductivity is directly proportional to the power absorp- tion, an additional heat ΔQ is added, thus leading to a rise in Te, tion coefficient of the material α = σ′/cnε0, where σ′ is the real part as illustrated in Figure 11a. of conductivity (see Equation (6)), n is the real part of refractive index, c is speed of light in vacuum, and ε0 is vacuum permit- Second, we need to impose a condition on electron density tivity. The excitation of THz current j in doped graphene thus conservation for the intraband THz excitation of graphene: at any electron temperature N c = ∫ 0 εD(ε ) f FD (ε , µ ,Te )dε = constant . ∞ facilitates the transfer of energy from the absorbed THz field to the free conductive electrons in graphene. In the approximation This requires that the rise in temperature Te due to the THz of sharp Fermi edge, which is convenient for this discussion, one deposited heat ΔQ must be accompanied by a reduction in can assume that this energy is initially confined to the electrically the chemical potential μ, as shown in Figure 11b,c (see also conductive states residing at the Fermi level EF. ref. [82]). The condition of carrier density conservation thus The linear energy–momentum dispersion in graphene given leads to the second thesis of the thermodynamic model: by Equation (2), apart from the high electron mobility due to (2) Carrier density conservation: Increase of electron tempera- fundamental absence of Coulomb binding, yields yet another ture leads to reduction of chemical potential in THz-excited unique property: perfect energy and momentum conservation graphene. for collinear electron–electron scattering. In a simple intui- tive picture, given available phase space, any two Dirac elec- Now that we established that the intraband THz absorption trons in graphene can scatter by conserving their total energy in graphene leads to an increase of electron temperature accom- and momentum: One electron increases its energy and panied by a downshift in the chemical potential, we arrive at a momentum by ΔE and Δk, respectively, while the second elec- qualitative understanding of the reduction of THz conductivity tron decreases its energy and momentum by the same amounts in THz-excited graphene, based on the spectral weight conserva- (see Figure 11a). Detailed considerations indeed showed that tion rule, illustrated in Figure 11d. This rule was first applied to collinear scattering between two graphene electrons is the the treatment of THz photoconductivity in graphene by Frenzel most probable, and therefore the fastest electron scattering et al.[132] Indeed, the increase of the electron temperature, resulting process.[98,99,139] This is in contrast to materials with quasi-par- in the downshift of the chemical potential µ, leads to unblocking abolic energy–momentum dispersion, such as semiconductors, of spectral density for previously Pauli-blocked interband transi- where typically more than just two participating particles are tions at the (infrared) photon energy of ħω > |2µ|. This increases needed in order to fully conserve both energy and momentum the spectral weight for interband transitions for graphene with in a scattering event, thus reducing its probability and conse- hotter electron population, whereas the total spectral weight of quently slowing the scattering rate down. both interband and intraband transitions must remain conserved. It is via these collinear electron–electron interactions in This brings us to the third thesis of the thermodynamic model. graphene, that the energy of the absorbed THz field, initially confined to the conductive states at the Fermi edge, is being redistributed across the entire carrier population of graphene, (3) Spectral weight conservation: The downshift of chemical becoming its collective kinetic energy—the electronic heat. potential in graphene with higher electron temperature leads The process of such an energy redistribution, the electron ther- to an increase of the spectral weight for interband transitions, malization, is known to happen on an ultrafast timescale of compensated by a decrease of the spectral weight for intra- <100 fs for both intra- and interband excitation of doped gra- band transitions. The latter reduces the THz conductivity in phene.[100,140–143] This thermalization is much faster than the graphene with higher electron temperature. oscillation cycle of THz waves, which is on the order of pico- Now we consider the dynamics of the processes underlying seconds. As a result, one can treat the thermalization dynamics the thermodynamic model of conduction in graphene. While as an instantaneous process on the timescale of THz–graphene the energy transfer from the driving THz field to the free interaction, which leads to the following thesis: ­electron population of graphene, raising the electronic temper- ature, occurs quasi-instantaneously on sub-100 fs timescale, the (1) Energy conservation: The THz energy absorbed by graphene consecutive electron cooling occurs on a slower, few-picosecond electrons via intraband conductivity is instantaneously con- timescale of phonon emission, as schematically depicted in verted into electronic heat with perfect conversion efficiency, Figure 11b. This dynamics of electron cooling after intraband raising the electronic temperature. excitation of doped graphene was measured in the pioneering Adv. Optical Mater. 2020, 8, 1900771 1900771 (16 of 25) © 2019 The Authors. Published by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.advancedsciencenews.com www.advopticalmat.de tr-ARPES experiments by Gierz et al.,[100] see Figure 12a. The tivity in graphene on the timescale of the oscillation cycle of the electron cooling dynamics features three distinct timescales THz field. Note how this dynamics of electron cooling, directly ranging from ≈10 fs to 1 ps, which are assigned to the electron observed in tr-ARPES measurements,[100] roughly corresponds relaxation via optical and acoustic phonons, respectively (see to the dynamics of THz conductivity recovery in a THz pump– also Table 1). The disparity between electron heating (sub-100 fs) THz probe experiments by Hwang et al.[54,55] (see Figure 5d). and total cooling (several ps) timescales naturally leads to the Further, the tr-ARPES data by Gierz et al.[100] demonstrated heat accumulation in the electronic population of THz-driven saturation of the electron temperature in graphene with graphene, ultimately resulting in the reduction of THz conduc- increasing intraband excitation fluence, see Figure 12b. Such Figure 12. Dynamics of electron heating and cooling under intraband excitation, and its effect on the THz conductivity and absorbance of graphene: a) Dynamics of electron heating and cooling in doped graphene under intraband excitation as measured in tr-ARPES experiment. b) Saturation of electronic temperature with increasing intraband excitation fluence (a,b: Reproduced with permission.[100] Copyright 2013, Springer Nature). c) Temporal evolution of electronic temperature in graphene with EF = 70 meV under quasi-single-cycle THz excitation of variable field strength (gray area indicates the modulus of driving THz field). d) Measured (symbols) and calculated (solid lines) nonlinear conductivity spectra of doped graphene, as the peak THz field increases from 2.3 kV cm−1 (yellow) to 120 kV cm−1 (black). e) Measured and calculated nonlinear absorption of doped graphene with EF = 70 meV, and the accumulated thermal energy density retained in the electronic system of graphene on the timescale of the interaction with quasi-single-cycle THz pulse, as well as the peak and average (in brackets) values of electronic temperatures as functions of THz peak field strength (arrows indicate relevant axis) (c–e: Reproduced under the terms of the Creative Commons CC-BY 4.0 License.[59] Copyright 2015, Springer Nature). Adv. Optical Mater. 2020, 8, 1900771 1900771 (17 of 25) © 2019 The Authors. Published by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.advancedsciencenews.com www.advopticalmat.de a saturation indicates a fundamental nonlinearity in the intra- Such a numerical propagation can be performed fully numeri- band excitation of graphene: raising the electron temperature cally as is customary in traditional nonlinear optics (see, e.g., suppresses the intraband conductivity, thus limiting the energy refs. [145,146] and references therein), or using convenient transfer rate for the following portion of the excitation signal. semi-analytical parameterization (see refs. [59,63] including the This saturation occurs on top of the sublinear relation between Supporting Information and Methods sections). the electron temperature and the incident power, which is In Figure 12c,d, the results of such a numerical propagation caused by the electron heat capacity increasing with electron of the experimentally measured incident THz pulse through temperature. Note again the similarity in the saturation of the CVD graphene with EF = 70 meV and τ(EF) = 140 fs are tr-ARPES—measured electronic temperature in the work by shown, assuming the long-range Coulomb scattering scenario Gierz et al.[100] (Figure 12b) and the saturable THz absorption for the hot electrons.[59] Figure 12c shows the calculated evo- in THz doped graphene from the pioneering work of Hwang lution of the electronic temperature in graphene, driven by et al.[54,55] (Figure 5c). near-single-cycle THz pulse of variable field strength, whose Now that both the mechanism of energy transfer from modulus is shown as a gray area. The evolution of electronic THz field to graphene— the intraband conductivity according temperature Te generally reproduces the dynamics of the THz to Equation (6), and its dynamics—quasi-instantaneous elec- field. However, on the timescale of the light–matter interaction tron heating and picosecond-timescale electron cooling (see Te never reaches the initial temperature of 300 K, since the total Figures 11b and 12a) are established, one can numerically ­ cool-off of the electronic population is a few-picosecond time- model the interaction of graphene with the driving THz fields scale process, which is only complete after the main part of the in the time domain. driving THz field has gone (see ref. [100] and Figure 12a). This We note here that the intraband conductivity given by results in permanent electronic heat accumulation in graphene Equation (6) also contains an electron energy-dependent scat- on the timescale of the interaction with the driving THz field, tering time τ(ε). For graphene in full thermal equilibrium, reducing the overall THz conductivity and absorbance of gra- that is, before strong THz excitation, this scattering time phene, as the THz field is transmitted through it. corresponds to the scattering time at the Fermi level τ(EF), The numerical transmission of the THz field through gra- which can be determined experimentally in a linear THz-TDS phene using the thermodynamic model yields the nonlinear experiment such as Figure 2f. Even though several electron THz spectra of graphene, shown in Figure 12d (solid lines) scattering regimes are postulated for (hot) electrons in gra- together with the experimental data (symbols), for the THz phene, there is no full consensus on this topic in the literature fields ranging from 2.3 to 120 kV cm−1.[59] The conductivity until now. The two dominating momentum scattering mecha- in THz-driven graphene decreases with both the THz field nisms are known to be the long-range Coulomb scattering and strength and the frequency. It is evident that the agreement short-range disorder scattering, for which the scattering times between the calculations and the entirety of the experimental can be approximated as, respectively, τ (ε) = γε and τ (ε) = β/ε, data is quantitative, lending credence to the theses underlying where γ and β are linear proportionality constants (see, e.g., the thermodynamic model of the nonlinear THz conduction of refs. [28,59] for discussion). Typically, long-range Coulomb scat- graphene.[59] tering with τ (ε) = γ ε is believed to dominate for CVD-grown Figure 12e shows the measured and the calculated power graphene deposited on a regular dielectric substrate (such as absorption of the graphene sample as function of the THz SiO2), whereas this scattering channel can be suppressed in field strength, demonstrating a very strong saturable absorp- exfoliated graphene that is suspended or encapsulated by hex- tion effect, as has also been observed in many previous meas- agonal boron nitride. Assuming that one of these scattering urements (see, e.g., refs. [55,56,58,59,111] and Section 4.2 for mechanisms dominates, the proportionality constant, whether discussion). The THz power absorption of doped graphene, γ or β, can thus be established experimentally from a linear measured here over the full timescale of THz–graphene inter- THz-TDS experiment at room temperature as γ = τ(EF)/EF and action of several picoseconds, reduces from over A = 0.125 β = τ(EF)EF, respectively. Both the single value for scattering down to almost zero as the driving THz field increases from time τ, and the Fermi energy EF can be readily determined in 2.3 to 120 kV cm−1. This effect of very strong THz intraband such a measurement (see Figure 2c–f and refs. [59,63,144]). absorption bleaching is in perfect agreement with the satura- Once the dominating scattering mechanism for hot elec- tion of electron temperature in doped graphene under intra- trons, and hence the energy dependence of the scattering time, band excitation, observed in tr-ARPES experiments,[100] which, is assumed, one directly applies the conditions of energy and as mentioned before, in turn limits the energy transfer rate electron density conservation to the interaction of the driving from the driving THz field to graphene electrons. In the same THz field with graphene. The experimentally measured figure, the calculated accumulated electron heat, as well as the ­incident THz field is then numerically propagated in the time peak and average (in brackets) electronic temperature of gra- domain through the graphene, which at each point in time is phene is presented. The energy balance shows that in this case characterized by instantaneous conductivity from Equation (6) about 15% of the absorbed THz energy remained within the dependent on instantaneous value of electron temperature electronic system of graphene on the timescale of interaction Te and chemical potential µ, and with the energy-dependent with the THz field, irrespective of the THz field strength. The scattering time τ(ε). This numerical propagation generates calculated electronic temperatures in 1000s of Kelvin range, the transmitted THz field that can be analyzed in the same reached as a result of THz excitation, are in agreement with fashion as in nonlinear THz-TDS experiments, yielding the the measurements of electronic temperature in tr-ARPES nonlinear, field-dependent conductivity spectra of graphene. experiments.[100] Adv. Optical Mater. 2020, 8, 1900771 1900771 (18 of 25) © 2019 The Authors. Published by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.advancedsciencenews.com www.advopticalmat.de As we already mentioned in Section 4.2.2, the thermody- at a defined frequency f, the induced nonlinear current in namic model of intraband electron dynamics in graphene is graphene will contain oscillations at higher order harmonics also applicable to the THz response of doped graphene under of the driving frequency. Due to the centrosymmetry of gra- optical interband excitation. The tr-ARPES experiments with phene, even-order harmonic generation is forbidden, and thus interband pump,[100,147,148] as well as optical pump–probe spec- only odd-order higher harmonics 3f, 5f, 7f, … etc., will be gen- troscopy[139,149] showed a similar effect of increase of electron erated. We note that the generation of high-order THz har- temperature in doped graphene subject to interband excitation. monics in graphene has long been proposed in theory (see, e.g., Indeed, also in the case of interband optical excitation the col- refs. [19,22,23]), yet the experimental demonstration of this linear electron–electron scattering of Dirac electrons will lead long-sought effect has been achieved only recently. to at least partial redistribution of the energy of photoexcited In 2018, Hafez et al.[63] have demonstrated extremely efficient electron–hole pairs to the background free carrier population, THz high-harmonics generation in graphene using the thermo- thus increasing its temperature and driving down THz con- dynamic response of its free electrons to driving quasi-mono- ductivity of graphene.[84,129,133] As a result, THz conductivity of chromatic THz fields, following the thermodynamic model by photoexcited graphene allows one to estimate the electronic tem- Mics et al.[59] The multicycle quasi-monochromatic THz fields perature, and hence the efficiency of electron heating induced with peak electric field up to 85 kV cm−1 at frequencies between by optical excitation. We note here that the electron heating 300 and 700 GHz, used as a pump signal, were generated by efficiency under optical interband excitation is not supposed to a superradiant THz undulator facility TELBE at Helmholtz- be perfect, unlike the THz-driven intraband heating efficiency. Zentrum Dresden-Rossendorf.[124] As a sample, a regular CVD- The reason is that the photoexcited electrons and holes usu- grown graphene deposited on a SiO2 substrate was used, with ally possess enough energy in order to emit optical phonons of the Fermi energy of EF = 170 meV, and the electron momentum ≈200 meV energy, and hence to transfer their energy directly to scattering time at the Fermi level of τ(EF) = 47 fs. the lattice of graphene, without exciting the background elec- The quasi-monochromatic multicycle THz driving of tron population. Optical phonon emission thus becomes the graphene significantly enhances its effective nonlinear con- competing process to the background electron heating in gra- ductivity response, as compared to the single-cycle excitation phene. In ref. [84], Jensen et al. used optical pump–THz probe demonstrated in refs. [55,56,58,59,111]. The key reason is that spectroscopy to estimate the optically driven electron heating the electron cooling time, which is on the order of picoseconds, efficiency in photoexcited doped graphene. Using a numerical conveniently matches the oscillation period of a multicycle version of the model by Song and Levitov, ref. [129] and its Sup- pump THz wave. This allows the graphene conductivity (and porting Information, the photoconductivity spectra of photo­ hence its absorbance) to efficiently recover in time for the next excited graphene were described, thus extracting the dependence electric field cycle to arrive. Such an efficient recovery of con- of the electronic temperature and the total thermal energy of ductivity of graphene, in turn, facilitates more efficient energy the electron gas in graphene on the fluence of photoexcitation. transfer from the THz pump wave to graphene electrons over It was shown that in the case when the total energy of the unex- the whole timescale of THz–graphene interaction, which for a cited electron gas (“depth of the Fermi sea”) was significantly multicycle THz pump wave can last 10s of picoseconds. lower than the absorbed energy of the interband photoexcitation Figure 13 shows the thermodynamic calculations, following of graphene, all the energy of the photoexcited electrons and the model by Mics et al.[59] of the THz-induced electron heat holes was converted into electron heat, without the direct exci- and the corresponding temporal modulation of graphene tation of the lattice via optical phonon emission. However, for absorbance (and hence its conductivity), driven by the multicycle higher excitation fluence and lower Fermi energy, the electron THz field oscillating at a fundamental frequency of 0.3 THz. heating efficiency was found to decrease at the expense of the This calculation particularly exemplifies the highly nonlinear direct coupling of photoexcited electrons to the lattice. Tomadin nature of response of free electrons in graphene to THz exci- et al.[133] further developed the findings of the work of ref. [84] tation: the driving field strengths of 10 and 85 kV cm−1 lead using a more advanced microscopic model. to dramatically different effects on the electron temperature in graphene, and its absorbance. If the effect of 10 kV cm−1 excita- tion is minor, leading only to a slight perturbation of the equi- 4.4. THz High-Harmonic Generation librium properties of graphene, the excitation with 85 kV/ cm−1 THz field strength results in a large THz-induced heat depo- As described in the previous section, the interaction of strong sition into the system, and significant absorption and hence THz field E(t) with the free electrons in graphene results in conductivity modulation featuring pronounced temporal strong suppression of graphene conductivity via carrier heating oscillations observable twice per optical cycle of the driving on sub-100 fs timescale, which is followed by the conductivity field. This is a particular consequence of the centrosymmetry recovery via carrier cooling by phonon emission, occurring of graphene, where both the positive and the negative THz field on the timescale of a few picoseconds (see Figure 11). As a oscillation half-cycle produces the same effect on its properties, result, during the interaction with the THz field, the instanta- in this case, suppressing its conductivity. neous conductivity of graphene σintra(E,t) becomes time- and Figure 14 shows the schematic of the experiment from the field-strength dependent. This conductivity nonlinearity in work of Hafez et al.,[63] the spectrum of the driving THz field at turn leads to the generation of nonlinear current in graphene the frequency f = 0.3 THz, as well as the spectrum of the THz j(t) = σintra(E,t)E(t) in response to the driving THz field E(t). If field transmitted through graphene sample, with clearly visible the driving THz field is monochromatic, that is, it oscillates higher odd-order THz harmonics up to the seventh order. The Adv. Optical Mater. 2020, 8, 1900771 1900771 (19 of 25) © 2019 The Authors. Published by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.advancedsciencenews.com www.advopticalmat.de only one atomic layer thick: They demonstrate the values slightly less than 10−2, 10−3, and 10−4, for the third, fifth, and seventh THz harmonics, respectively. The total nonlinear field conversion effi- ciency therefore tends to ≈1%. The spectral range in this experi- ment was limited by the detector bandwidth, however, the signal- to-noise ratio permitted by the TELBE THz source and the detec- tion scheme[124] was not a limiting factor in this measurement. In Figure 15a,b, the measured HHG signals in the time domain, as well as the corresponding results of the thermody- namic model calculations using the experimental pump wave are presented. The agreement between the measurement and the calculation is rather good: The thermodynamic calcula- tion captures well all the key features of experimental signals, including their magnitude, distortion, and the shortening of the generated harmonic pulses with increase in their order. Figure 15c summarizes the experimental result of THz HHG in graphene by showing the electric field strength of the gener- ated third, fifth, and seventh harmonics, as a function of the pump field strength (symbols). In the same figure, the results of the thermodynamic model calculation are shown, using the Figure 13. THz field-induced heating and absorption modulation: experimental pump field and basic parameters of graphene a) Multicycle quasi-monochromatic THz driving field oscillating at in full thermal equilibrium at 300 K as an input. The agree- 0.3 THz, b,c) the temporal THz induced electronic heat density (energy ment between the data and the model is quantitative. From the per unit area), and the THz power absorbance, respectively, of graphene observed dependencies in the regime of small-signal nonlin- driven by the intense multicycle THz pulse in (a) at two THz peak field earities (pure power law, dashed lines), one can now estimate levels of 10 kV cm−1 (the blue curves) and 85 kV cm−1 (the red curves), demonstrating the thermal response of the Dirac electrons of doped the effective nonlinear optical coefficients of graphene up to the graphene to the intense THz driving field (Reproduced with permission.[63] seventh order: χ(3)eff ≈ 1.7 × 10−9 m2 V−2, χ(5)eff ≈ 1.2 × 10−22 m4 V−4, Copyright 2018, Springer Nature). and χ(7)eff ≈ 1.74 × 10−38 m6 V−6 (see ref. [63] and its Methods section for details). These values are extremely large, and by fundamental field at f = 0.3 THz is transmitted through the gra- many orders of magnitude exceed the nonlinear coefficients of phene with the transmission coefficient exceeding 0.9, which any other known electronic material. This possibly makes gra- serves as a convenient reference for the conversion efficiency phene the most nonlinear material known to date. from the THz pump field to higher harmonics. These c­ onversion Now that the thermodynamic model of THz response of efficiencies are remarkably high, especially for a material which is graphene has demonstrated its high accuracy in reproducing Figure 14. THz high-harmonic generation in single-layer graphene up to the seventh order at room temperature: a) schematic of the THz odd-order harmonics re-emitted from graphene driven by a multicycle THz field, b) experimental data of the THz field spectrum transmitted through the graphene sample (in blue) showing odd harmonics up to the seventh order relative to the spectrum of the driving field at a fundamental frequency of 0.3 THz (in red) (Reproduced with permission.[63] Copyright 2018, Springer Nature). Adv. Optical Mater. 2020, 8, 1900771 1900771 (20 of 25) © 2019 The Authors. Published by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.advancedsciencenews.com www.advopticalmat.de Figure 15. THz high-harmonic generation in single-layer graphene up to the seventh order at room temperature—experiment versus calculations: a,b) the temporal THz fields of the fundamental driving field and the generated harmonics; experiment in (a) and calculations in (b). c) The depend- ence of the generated higher-harmonic field on the THz driving field; symbols for the experiment, solid black lines for the calculations, and dashed lines for the power-law fit of the low-signal nonlinearity regime below a driving field strength of 20 kV cm−1 (Reproduced with permission.[63] Copyright 2018, Springer Nature). the THz nonlinearity in graphene to single-cycle[59] and be observed at the fields as weak as only 10s of kV cm−1, which multicycle[63] excitation, one can use it to predict the generation is 2–3 orders of magnitude weaker than that typically required of THz harmonics of even higher orders. In Figure 16, the cal- for initiation of the coherent Bloch oscillation mechanism. culated spectra of response of graphene to the excitation at the fundamental frequency f = 0.68 THz are shown, predicting the appearance of harmonics up to the 13th order within 5. Conclusions and Future Perspectives the dynamic range of 106, characteristic of the TELBE THz facility. The driving field amplitude in this calculation had a In this Progress Report, we have described the birth and evolu- maximum of 1 MV cm−1, which is a conservative estimate of tion of the research area of THz nonlinear optics of graphene. the field strength required to enter the ballistic nonlinearity We have shown that the strongest nonlinear response in gra- regime of coherent Bloch oscillations (see estimate provided in phene is facilitated via an intraband conductivity channel, and Section 4.1). We note here again, that the HHG output via the that the underlying physical mechanism is a very powerful ther- collective thermodynamic response of graphene electrons can modynamic, collective response of its free carrier population to the driving THz fields. We have summarized the fundamental aspects of a thermodynamic model of the THz nonlinearity of graphene, first described in the work by Mics et al.,[59] which rests on basic conservation laws, and relies on perfect conver- sion of the THz energy absorbed via the intraband conductivity mechanism into electronic heat of the free electron popula- tion of graphene. This mechanism explains and quantitatively reproduces the experimentally observed strong THz saturable absorption and nonlinear wave conversion, including extremely efficient THz high harmonics generation in graphene at room temperature and under ambient conditions. As we show, gra- phene at THz frequencies possesses possibly the highest non- linear coefficients of all materials we know to date. We furthermore show that the thermodynamic mechanism underlying the enormous THz nonlinearity in graphene is active at rather moderate driving fields of only 10s of kV cm−1. This THz field strength is an order of magnitude lower than the typical channel field of modern high-speed field-effect transistors, Figure 16. Thermodynamic model calculations of THz high-harmonic routinely operating at 100s of GHz frequencies. This is of high generation up to the 13th order. Calculated spectrum showing high har- technological relevance since graphene is easily compatible monics up to the 13th order within a dynamic range of 106 with respect to the transmitted field spectrum at the fundamental frequency (considering with electronic technologies such as CMOS.[150,151] This creates a fundamental frequency of 0.68 THz) (Reproduced with permission.[63] the possibility to utilize CMOS-based sub-THz electronics as Copyright 2018, Springer Nature). efficient and cost-effective pump sources for hybrid devices for Adv. Optical Mater. 2020, 8, 1900771 1900771 (21 of 25) © 2019 The Authors. Published by WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim www.advancedsciencenews.com www.advopticalmat.de THz frequency synthesis in graphene via the thermodynamic [2] K. S. Novoselov, A. K. Geim, S. V Morozov, D. Jiang, HHG mechanism. We further note that the observed extremely M. I. Katsnelson, I. V Grigorieva, S. V Dubonos, A. A. 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