Hot charge carriers in graphene exhibit fascinating physical phenomena, whose understanding has improved greatly over the past decade. They have distinctly different physical properties compared to, for example, hot carriers in conventional metals. This is predominantly the result of graphene's linear energy–momentum dispersion, its phonon properties, its all-interface character, and the tunability of its carrier density down to very small values, and from electron- to hole-doping. Since a few years, we have witnessed an increasing interest in technological applications enabled by hot carriers in graphene. Of particular interest are optical and optoelectronic applications, where hot carriers are used to detect (photodetection), convert (nonlinear photonics), or emit (luminescence) light. Graphene-enabled systems in these application areas could find widespread use and have a disruptive impact, for example in the field of data communication, high-frequency electronics, and industrial quality control. The aim of this review is to provide an overview of the most relevant physics and working principles that are relevant for applications exploiting hot carriers in graphene. Hot charge carriers in graphene exhibit fascinating physical phenomena and have great promise for exciting optoelectronic applications. The current understanding of the relevant fundamental physics and the most promising applications are reviewed. Understanding and controlling the properties of hot carriers – electrons and holes with excess kinetic energy – is a paradigmatic topic in both physics and chemistry. In graphene, hot carriers are particularly consequential because they can be efficiently created, controlled and exploited towards applications, as we will discuss in detail in this review. Hot carriers in graphene have been addressed experimentallyIn this review we will focus on the aspects of hot carriers that are most relevant for applications involving light, and on systems where graphene is the active material containing hot carriers. Our aim is not to provide a fully exhaustive review of all relevant literature in this expansive and expanding field. We refer to several reviews with relevant background information: ref. Te is higher than the (equilibrium) temperature of the lattice TL.We distinguish two types of hot carriers, where in both cases charge carriers have an increased kinetic energy. We will use the term “high-energy charge carriers” for those that have an increased energy, yet do not form part of a thermalized carrier distribution. In this case, it is not possible to define a meaningful temperature, even if the carrier energy can be expressed in units of temperature. We will exclusively use the term “hot carriers” for those high-energy carriers that have thermalized with other charge carriers, thus forming a Fermi–Dirac distribution with an increased temperature – a hot-carrier distribution see . In thisTe, assuming that the electron density remains constant over time t:Ce is the electronic heat capacity. The first term on the right, Q̇ext, is the rate at which heat is added by external sources and removed via internal or external cooling channels, for example via electron–phonon collisions. The other two terms on the right-hand side are related to transport phenomena. The second term corresponds to Fourier's law of heat conduction, and contains the electronic part of the thermal conductivity κe. The third term describes the energy carried by the electric current density. In this term, Π is the local Peltier coefficient of the material, V is the local voltage, andσ the conductivity. The Peltier coefficient is related to the local Seebeck coefficient of the material via the Onsager reciprocity relation S = Π/Te. We will discuss the different terms of As a starting point for describing the relevant physics of hot carriers in graphene, we provide the thermoelectric equation describing the time evolution of the electronic temperature Excitation, we will discuss different approaches for exciting graphene such that carriers are heated, followed by section 3. Energy dynamics on the thermalization and cooling dynamics of the electronic system. These two sections are mainly related to the term Q̇ext of Conductivity, we will discuss the linear and nonlinear conductivity of hot carriers. We will then discuss transport phenomena in section 5. Transport, in particular diffusion and thermoelectric effects, related to the last two terms of Applications, followed by section 7. Discussion and outlook. Finally, we conclude with section 8. Conclusion. We also provide two boxes with background information: Box 1. Boltzmann theory and Box 2. Experimental techniques.This review is organized as follows: In section 2. 2In this section we will describe different ways of exciting graphene, such that hot carriers are created. We then discuss, from a macroscopic thermodynamic viewpoint, how a hot-carrier distribution is established after excitation.2.12.1.1via electron–phonon scattering, and leads to a steady-state condition with equal carrier temperature Te and lattice temperature TL in the heated region. The most common way to heat up the graphene lattice locally is by using micro-heaters in the proximity of the graphene sample18,19 , where Pin is the absorbed power density of the external source,  is the heat transfer coefficient in units W m−2 K−1, and τcool is the cooling time. The heat transfer coefficient describes the flow of heat out of the electronic system, for example through electron–phonon coupling. The thermoelectric equation, Steady-state heating refers to the situation where graphene is excited either continuously, for example with CW light or a DC current, or by a time-dependent power density input that varies on a timescale that is much slower than the heating–cooling dynamics of graphene charge carriers. Under these excitation conditions, a new equilibrium situation will be established with an increased carrier temperature Te, for a given heating power density Pin, it is beneficial to operate with a small electronic heat capacity and a long cooling time. We will discuss the cooling dynamics in the next section. The electronic heat capacity, or specific heat, in two limiting cases – “doped” and “undoped” – is given by:29,30In order to increase the carrier temperature ħ is the reduced Planck constant, vF the Fermi velocity, kB Boltzmann's constant, and ζ(3) ≃ 1.202. In the “doped” regime, the heat capacity decreases upon decreasing EF, as there are fewer electrons to share the energy. From the perspective of the specific heat, it is thus advantageous to operate at low Fermi energy, if the objective is to maximize the temperate increase ΔTe. Upon approaching the Dirac point, the EF-dependence drops out. In this case, ΔTe can be further maximized by operating at low temperatures. We note that graphene can reach record-low values of the electronic heat capacity, thus leading to large temperature rises for small excitation. Heat capacities as low as 2 × 10−9 J (m2 K)−1,31 and very recently even 1.2 × 10−11 J (m2 K)−1,32 have been found using noise thermometry measurements.Here, 2.2.2τtherm to this process. If τtherm is shorter than the cooling time τcool, a quasi-equilibrium situation is established after absorption of the pulse and thermalization. In this case, we can use the peak energy density, or input fluence,  in order to calculate the peak increase in electron temperature ΔTe,peak. Here, frep is the repetition rate of the incident excitation, and the input fluence is the absorbed fluence (not the incident fluence). A lower repetition rate gives rise to a larger ΔTe,peak. In order to obtain the peak temperature, we solve the integral , and obtain the following sub-linear relationships between peak temperature and input fluence:Transient heating refers to the situation where graphene is excited by a short pulse that varies on a timescale faster than the heating–cooling dynamics. In the simplified case of an infinitely short input pulse of power, absorption is followed by thermalization, where a common temperature of the electronic system is established. We can assign a thermalization time 2.2.3η. One then replaces Fin by ηFin, as done for example in ref. −2 range, a photoluminescence study found a significantly lower Te than expected, suggesting a rather low heating efficiency.30 On the other hand, optical pump – terahertz probe studies with an absorbed fluence in the nJ cm−2 range found heating efficiencies well above 50%,34,35 demonstrating that thermalization within the electronic system dominates over alternative energy relaxation channels. Below, we will discuss the microscopic processes that govern the heating efficiency.If the thermalization process – where high-energy carriers interact with other carriers to form a hot-carrier distribution – competes with other energy relaxation processes see , it can 3In this section, we describe the microscopic processes governing the energy dynamics of hot carriers. We start with a thermodynamic picture of carrier heating, followed by a description of the carrier–carrier interactions that lead to thermalization of the electronic system, after having absorbed a heating pulse, for example by photoexcitation. During these carrier–carrier interactions the total energy is conserved, and the electronic system typically evolves from a non-thermal to a thermal distribution with elevated carrier temperature. We then discuss the different interactions between carriers and other (quasi-)particles that lead to cooling of the hot-carrier system by damping of heat to a bath. Theoretical details on calculating these scattering events can be found in Box 164,65 in which electronic states are described in terms of wave packets.66,67 When the length scales over which external electric (E) and magnetic (B) fields vary are much larger than the spread of wave packets themselves,66 the latter can be treated as point-like particles with well-defined position r (the “centre” of the wave packet) and wavevector k. Then, as usual in statistical mechanics, the properties of a large collection of particles can be determined by how they arrange in the phase space. The central object becomes the distribution function, f, which determines at every instant of time t the number of particles in an infinitesimal phase-space volume element centered around the point . In general, the number of particles therein changes over time. Firstly, when external forces are applied, after an infinitesimal time dt, the particles’ position and wavevector become r + ṙdt and k + Fdt/ħ, respectively.66,67 Here, F = −e(E + ṙ × B), while ṙ is the particle velocity which, barring Berry-curvature corrections,67 is equal to ħ−1∇kεk, where εk is their energy dispersion.The statistical modelling of thermodynamic out-of-equilibrium systems is usually performed by means of a “microscopic” Boltzmann equation, from which via the collision integral64I[f]. The collision integral accounts for all many-body scattering events a particle undergoes  that lead to a change of its wavevector k. After any such scattering event, a given particle transitions to a different phase-space volume element. Therefore, I[f] is responsible for the time evolution of the occupation function due to collisions.The number of particles in a phase-space volume element changes over time also because of collisions, described 64,65The time evolution of the distribution function, accounting for both external forces and collisions, is described by the following equation:via the Chapman–Enskog method.68 Introducing the electron heat capacity and other transport coefficients, the resulting equation can be recast in the form of 1.via emission of phonons and can be effectively described viaFinally, we note that the equation above can be used to describe also the thermalization of non-equilibrium electrons (see section 3.1), not just their subsequent macroscopic cooling dynamics  = 2ε/(πħ2νF2) is the density of states of graphene.36 When a system at equilibrium, with , is excited by an absorbed energy density Fin, energy conservation dictates that that is characterized by “hot” Fermi–Dirac distributions with an increased carrier temperature. The electrons and holes thus have Fermi–Dirac distributions that are broadened. This broadening has been observed using time-resolved ARPES measurements, after pulsed optical heating37–39  to the UV (ħω ∼ eV).Time-resolved ARPES. In an angle-resolved photoemission spectroscopy (ARPES) experiment, high energy photons  impinge on a crystal in vacuum, ejecting electrons in free space, where their energies and exit angles are measured. These are related to the energy and crystal momentum of the electrons inside the sample. In the case of time-resolved ARPES, the sample is first excited by an ultrashort light pulse which generates high-energy charge carriers and subsequently the ARPES spectrum is measured with a second delayed high photon energy ultrashort pulse.111Time-resolved photocurrent. In time-resolved photocurrent (trPC) experiments the sample is excited by two ultrashort laser pulses and the electrical response of the sample (photocurrent or photovoltage) is measured as a function of their delay. If the electrical response scales nonlinearly with the incident laser intensity, such as any hot-electron dominated photocurrent/photovoltage in graphene, the signal at zero/small delays will differ from the signal at large delays, and the recovery dynamics of the signal will reflect the time response of the sample.33,53,112–114nhot – the density of carriers with an energy larger than the chemical potential, given by††private communication.A. Tomadin, It will be useful to consider the hot-carrier density μe = μh = 0, this reduces to the thermal carrier density ntherm = (π/6)(kTe/ħνFB)2, often used for graphene under electrical heating.40,41 Whereas the carrier density ne − nh is conserved during thermalization, the hot-carrier density nhot increases. Several studies16,34,35,42–44 have discussed this effect quantitatively. In the next subsection, we will discuss the microscopic processes that are responsible for this.For intrinsic graphene with kBTe ≪ EF, only the carrier density of one band needs to be considered for energy and particle conservation. Thus, in the case of electron doping, it is enough to consider conservation of ne, because nh ≈ 0. The situation is different closer to the Dirac point, where kBTe ≫ EF. This is because the conserved quantity ne − nh allows for additional electron–hole pair creation across the bands, at least from a thermodynamic viewpoint. We illustrate this following ref. Pin, yet before thermalization, and 300 fs later, when thermalization has occurred. These results are based on microscopic charge interactions , this broadening only affects the electrons in the conduction band. The density of holes in the valence band was basically zero before thermalization, and is still zero after thermalization. The density of electrons in the conduction band stays constant, whereas the density of hot electrons in the conduction band nhot,e has clearly increased. Therefore, thermalization in the case of large EF is referred to as intraband thermalization. In the case where EF is initially close to the Dirac point (EF = 0.05 eV), the broadening affects both electrons in the conduction band and holes in the valence band. Indeed, before thermalization there was a small density of holes in the valence band, whereas this density is increased after thermalization, as a result of charge and energy redistribution between the two bands. Also the density of electrons in the conduction band has increased. This means that – besides nhot – the density of interband electron–hole pairs, i.e. both ne and nh, has increased. Therefore, thermalization in the case of small EF is referred to as interband thermalization.We now discuss the creation of an increased hot-carrier density in the two doping regimes. In the “doped” regime, with 39,45 A time-resolved ARPES study37 furthermore showed a situation where the carrier distribution consists of two separate Fermi–Dirac distributions – one for electrons and one for holes. This so-called “inverted” state was found to have a lifetime of ∼130 fs. Interestingly, the short-lived situation of two separate distributions was only observed in the case of interband photoexcitation (with photon energy ħω > 2EF), whereas intraband photoexcitation (with photon energy ħω < 2EF) directly led to a single, broadened Fermi–Dirac distribution for the valence and conduction bands. These results suggest the occurrence of a rich interplay of dynamical processes during carrier thermalization. We will discuss these dynamics in the following sections.The thermodynamic picture above suggests that the valence and conduction bands are thermalized with each other, such that the holes in the valence band and electrons in the conduction band have a common chemical potential and carrier temperature. The simulations of ref. 3.1.2.via Coulomb carrier–carrier scattering events, where carriers exchange energy and momentum by keeping their total energy and momentum constant. We first classify carrier–carrier scattering processes into scattering events within a single band, and scattering events across the bands  high-energy carrier can create multiple secondary hot carriers during its cascade, transferring an average amount of Δε ≈ EF to each secondary hot carrier. This corresponds to the creation of multiple hot carriers per absorbed photon, gradually increasing nhot. The occurrence of hot-carrier multiplication was predicted and addressed quantitatively using optical-pump terahertz probe measurements with varying pump photon energy,34,42 and using time-resolved ARPES measurements.43 It was furthermore experimentally confirmed when Wu et al. succeeded in electrically collecting multiple hot-carriers per absorbed photon in a photodetector system.44 The occurrence of hot-carrier multiplication shows that carrier heating in graphene is an exceptionally efficient process that can be exploited for various applications (see section 6).In order to examine thermalization  2EF see . This crkBTe ≫ EF. In this regime interband thermalization occurs, and multiple free carriers can be created per absorbed photon. Winzer et al. showed that there is a competition between the interband processes of impact ionization and Auger recombination, and predicted the occurrence of carrier multiplication for relatively weak photoexcitation.50 Carrier multiplication was observed experimentally in ref. 35,46 , such as time-resolved photoluminescence30 and time-resolved ARPES37,38 typically found relatively low heating efficiencies. Microscopically, this is related to the relatively large density of initial high-energy carriers, and relatively high carrier temperature that is being established during thermalization. This slows down the thermalization process, thus making τheat longer. This means that the heating efficiency η becomes smaller, due to competing energy relaxation processes for high-energy carriers, with relaxation time τrelax .When keeping the photon energy fixed, while increasing the incident power, or fluence, the heating efficiency elax see . Furtherφ in 42,45,47,49,50,58 carrier–carrier scattering occurs preferentially in the near-collinear direction, due to the kinematic constraints of carrier–carrier scattering processes  constant, whose value has been determined using first principles calculations and Raman measurements.70–72 According to calculations in ref. τOP well below a picosecond. Scattering of high-energy carriers with optical phonons can therefore compete with Coulomb carrier–carrier interactions that lead to thermalization, and is therefore the main process determining the heating efficiency η that governs which fraction of the absorbed power density Pin ends up in the electronic system , followed by few-picosecond decay that was attributed to cooling of the hot optical phonons to acoustic phonons (τOP–AP), as the hot phonons are not able to cool fast enough, and re-heat the electronic system. This effect of the hot optical phonons is sometimes referred to as the hot-phonon bottleneck for carrier cooling. The optical phonon lifetime τOP–AP has been measured independently, for example through time-resolved Raman measurements, yielding values ranging from 1.2 to 2.55 ps.30,74,79–81Time-resolved studies using relatively strong excitation, leading to many high-energy carriers, typically observed ultrafast (few hundred fs) emission of so-called strongly-coupled optical phonons (SCOPs), rved see : sub-picε + μ) < EOP, the timescale of optical phonon emission increases exponentially  there is a significant fraction of carriers in the tail of the Fermi–Dirac distribution with a kinetic energy that is large enough to couple to optical phonons. Once these high-energy carriers have relaxed by emitting optical phonons, re-thermalization of the electronic system occurs. This leads to “newly excited” carriers with high enough energy to relax by emitting optical phonons. These processes of optical phonon emission and re-thermalization occur simultaneously. Thus, cooling occurs by the combination of optical phonon emission and continuous re-thermalization of the carrier system. A numerical simulation based on this intrinsic cooling mechanism was shown to be consistent with temperature-dependent cooling dynamics obtained with optical-pump THz-probe measurements.83 More recently, it was shown that this cooling mechanisms leads to bi-exponential decay, with a sub-picosecond initial decay related to direct coupling to optical phonons, and a few-picosecond decay due to the hot-phonon bottleneck.84 This study showed that a larger peak temperature and smaller EF typically give rise to slower cooling dynamics for this cooling channel. For graphene where alternative cooling channels (see below) are suppressed, this cooling channel ultimately determines the intrinsic limit of the hot-carrier lifetime of high-quality graphene (with a mobility >10 000 cm2 V−1 s−1) at room temperature.84For carriers with lower energy,  lead to scattering with phonons close to the centre of their Brillouin zone. There, acoustic phonons have nearly zero energy, and therefore cooling requires many successive interactions. This limitation, however, can be overcome through disorder-assisted scattering, as shown in ref. τSC at room temperature, especially in graphene with a sufficiently large disorder density, and associated low charge mobility. The occurrence of this cooling mechanism was shown experimentally using several experimental techniques shortly after its prediction,33,88–90 as shown in EF typically give rise to slower cooling dynamics for disorder-assisted supercollision cooling to acoustic phonons.Charge carriers with a kinetic energy that is not high enough to couple to optical phonons can couple to acoustic phonons. An important parameter governing acoustic phonon emission is the deformation potential, whose value is obtained from first principle calculations, 3.2.3.τsub. Low et al. theoretically addressed this cooling pathway in ref. 2, than in non-polar substrates, such as diamond. This is because of scattering with the surface polar phonon modes. Later, it was shown that if a substrate hosts hyperbolic phonon modes, the coupling between graphene hot carriers and these substrate modes can be even stronger, leading to picosecond cooling times at room temperature22,91,92  and transverse (TO) optical phonon frequencies.98 Owing to these two spectral regions with hyperbolic phonon polaritons, two optoelectronic experiments have shown that cooling of hot carriers in hBN-encapsulated graphene occurs via out-of-plane coupling to hyperbolic phonon polaritons in hBN.22,91,92 These studies showed that a higher peak temperature and a smaller EF typically give rise to slower cooling dynamics for super-Planckian cooling to hyperbolic substrate phonon modes . This cooling channel was observed using noise thermometry measurements, and – for their specific experiment – dominated for a lattice temperature below 80 K.99 It's important to note that this is a cooling process that is not captured by the term Γcool in The cooling pathways 100 and was suggested to explain the short-lived inverted state observed in ref. 101 Furthermore, hot-carrier photoluminescence – Planck radiation from the hot-carrier system – has been shown to occur.30 However, this process is inefficient: it gives rise to a Γcool of ∼10 W m−2 K−1,92 corresponding to a τcool of ∼10 ns. A theoretical proposal suggested that exploiting plasmonic near-field effects between neighboring graphene nano-islands could speed radiative heat transfer up to the femtosecond regime.102 A final cooling mechanism that was identified corresponds to resonant dissipation from individual atomic defects in graphene, as observed in ref. 63We briefly mention alternative cooling channels. Coupling to graphene plasmons could play a role in the early stages of the energy cascade,4σ governs both the electrical transport and the optical properties of graphene.115 In the following we will discuss the linear and nonlinear frequency- and temperature-dependent conductivities.The conductivity 4.1115–123 and experimental124,125,126,127–130 works dealing with the linear frequency-dependent optical conductivity of graphene σ(ω). Here we will focus on how the interband and intraband transitions of graphene depend on carrier temperature. Typically, for frequencies up to a few THz the intraband conductivity dominates, while for higher frequencies both contributions can play a role. Note that for graphene on a substrate the absorbance  is related to σ(ω) by , where  is the vacuum impedance, ns the substrate's refractive index and T is the transmission.130,131 This means that σ(ω) determines the number of photogenerated hot electrons, as it determines the optical absorption, and it's directly related to Joule heating as it determines the resistance of a graphene sheet (see section 2).There is nowadays a large amount of theoretical4.1.1125,126,132Graphene's interband optical conductivity can be written as:ħω ≫ 2|μ| and ħω ≫ 4kBTe, interband transitions dominate graphene's absorbance and σ(ω)inter = σ0 = e2/(4ħ), which thus becomes independent of frequency.124,125,133 For this value of the sheet conductivity the absorbance for isolated (suspended) graphene (ns = 1) depends only on elementary constants . Note that  is the fine structure constant. The same “universal conductivity” can be obtained from the Fermi golden rule for two-dimensional Dirac fermions.124For large enough photon energies, when 121,127 Although theory predicts that the approximation of a linear dispersion should hold only for relatively small values of the incident photon energy (ħω < 1 eV),124 experiments observed very small deviations from the universal value of the absorbance A ∼ 2.3% for photon energies up to ħω = 1.5–2 eV.124,127 For higher photon energies, the conductivity deviates significantly from the universal value.For large energies, graphene's band structure deviates from linear dispersion, and excitonic effects at the saddle point singularity at ∼5 eV lead to increased absorption.σ(ω)inter is that the value 2μ, which defines the crossing from intraband to interband transitions, can be tuned by doping and external gate voltages,134 leading to the possibility of designing gate-tuneable optical modulators.135 According to μ, interband absorption can either increase (for ħω < 2EF) or decrease (for ħω > 2EF), as shown in Another key aspect of 4.1.2123The intraband  conductivity for electrons or holes is given by:cf. ref. There are three main approaches that have been used to understand how this intraband conductivity changes upon carrier heating, which is less straightforward than in the case of interband transitions. Experimentally, this issue has been addressed in particular using optical pump – terahertz probe studies, 131,137In the first approach, the temperature dependence is fully captured by the so-called Drude weight:137 which is related to the Thomas–Reiche–Kuhn sum rule for light–matter interaction.140 For the graphene conductivity, this sum rule indicates that the broadening of the interband conductivity , which is in agreement with several experiments34,35,57,136–139 , as observed in ref. umr on Te was inserted phenomenologically, assuming Te = TL and momentum relaxation dominated by phonon scattering.This result comes from considering spectral weight conservation,Te/TF is small. Here, the energy-dependence of the momentum relaxation time was taken into account, under the realistic assumption that it is dominated by long-range Coulomb scattering with impurities. This corresponds to a momentum relaxation time that typically increases linearly with carrier energy,123 and is typically the channel that limits charge mobility at room temperature in substrate-supported graphene. This approach correctly gives the negative photoconductivity for EF away from the Dirac point, as observed experimentally.34,35,57,136–139 However, it can not be used close to the Dirac point, as the conditions for the Sommerfeld expansion are not met.In ref. τmr. This approach gives positive photoconductivity close to the Dirac point and negative photoconductivity away from the Dirac point, in agreement with experiments.35,57,137,139 According to ref. 35Finally, in ref. 4.1.3120,122,141–144 and experimentally. Of particular interest are the observation of high-harmonics generation (HHG),56,145 third-harmonic generation (THG),146–150 four-wave mixing (FWM)151–153 and different kinds of intensity-induced changes of the refractive index, such as saturable absorption (SA)154,155 and the Kerr effect.156 Being a centrosymmetric crystal, graphene does not display intrinsic even-order nonlinearities in the dipole approximation. For this reason, second-harmonic generation has been observed only due to extrinsic factors, such as breaking of symmetry at an interface157 and electric currents/fields,158,159 or at large incidence angles due to the quadrupole nonlinear response.160 An intuitive picture of the strong nonlinear optical response of graphene can be obtained from a semi-classical model considering the electric field induced sheet current density within the Dirac cone.161–163 An overview of the nonlinear optical properties of graphene and related layered materials can be found in ref. The nonlinear optical properties of graphene have been investigated extensively, both theoretically4.1.4σx,x where x is any in-plane Cartesian direction of the graphene sheet and E0 the incoming electric field responsible for SA.164 Note that SA is possible for both intraband and interband transitions143 and in both cases the dynamics of the effect is dominated by the heating and cooling dynamics of hot electrons. A sketch of the SA mechanism in the case of interband transitions is depicted in 154 The photo-excited hot electrons can inhibit optical transitions by Pauli blocking in a range of kBTe around EF, thus reducing the absorption of photons at energy ħω ∼ kBTe. In the low-excitation regime when kBTe is smaller than EF or the excitation photon energy, the (linear) absorption is independent of the excitation intensity. However, for increasing excitation intensity and higher kBTe hot electrons will induce Pauli blocking of interband transitions, leading to a nonlinear intensity-dependent saturable absorption.143,154 In the case of intraband transitions, SA is due to the photoinduced negative photoconductivity discussed above: an increase of the excitation intensity leads to a reduction of σintra |2, THG in graphene is fully captured by the nonlinear optical sheet conductivity |σ(3)|2, which can be microscopically calculated at Te = 0 K.141,164,165 Following symmetry considerations, the third-order optical sheet conductivity at Te = 0 K, which is in general a rank 4 tensor, can be reduced to a single in-plane element:141,148σ(3)0 = 4e4ħνF2/(32π) and G = ln[(x + y)/(x − y)]. As discussed, σ(3) has three sharp resonances at ħωm = 3EF with m = 1, 2 and 3 corresponding to one-, two- and three-photon vertical transitions  for different values of Te between 0 K and 1800 K: for increasing Te the multi-photon resonances, which are clearly visible at Te = 0 K, broaden until they merge and the modulation depth of the THG efficiency decreases dramatically. This can be qualitatively understood considering the Te-induced broadening of the Fermi–Dirac distribution. Since THG is a parametric process and the emission occurs only during interaction of the electric field with the nonlinear material, all nonlinear optical experiments on graphene that use ultrashort pulses (<100 fs) should always carefully consider the effect of a high Te. Notably, this is true also when ħω〈2|EF| due to broadening of the interband transitions at high Te and possible intraband transitions mediated by defects and phonons. The increase of Te is also responsible for large deviations from the typical cubic power dependence of THG experiments, as shown in ref. Te were observed also in FWM experiments149,153,166 and recently ultrafast all-optical modulation of THG with modulation depth above 90% was obtained in single-layer graphene.163The efficiency of nonlinear processes such as third-harmonic generation (THG) is strongly enhanced due to resonant transitions. This is true also for graphene, where the THG efficiency can be efficiently tuned by more than one order of magnitude when in resonance with vertical multi-photon transitions within the Dirac cone148,149 . In addinsitions .148,149 4.1.6122,161,162,167–169 However, experiments either did not detect any THz HHG from multi-layer graphene,170 or only observed weak signatures of THz harmonics at 50 K.171 These results were explained as the consequence of ultrafast e–e scattering (<100 fs) which suppresses the coherence in the THz-induced electron velocity and current density in materials with linear dispersion.172The case of THz high-harmonic generation (HHG) clearly illustrates the importance of hot electrons in the nonlinear optical response of graphene. Several theoretical works predicted the existence of an intense nonlinear response of graphene at THz frequencies, based on a nonlinearity mechanism relying on coherent electron motion.−3, 10−4 and 10−5 for third, fifth and seventh harmonics, respectively, was demonstrated at room temperature using doped single-layer graphene.56 The result was successfully interpreted based on a nonlinearity mechanism that does not rely on coherent electron motion. Rather, the THz nonlinearity was understood from a purely thermodynamic picture of the THz response of doped graphene. In this thermodynamic picture,56,173 hot carriers play a central role, because the intense nonlinear response of graphene is due to the combination of the THz-induced carrier heating that reduces the THz conductivity,54 together with the ultrafast heating and cooling dynamics of hot electrons  with Dirac fermions.175In the future, THz nonlinear optics with graphene could offer a rich playground for on-chip signal processing5In this section, we address transport properties related to hot carriers. We first discuss intrinsic properties such as hot-carrier diffusion and thermoelectric effects. Subsequently, we discuss interlayer phenomena, involving transport of graphene hot carriers to a neighboring material.5.15.1.1κe is the electronic part of the thermal conductivity. This equation takes the form of a standard diffusion equation with a thermal diffusivity , typically given in units of cm2 s−1. Since hot charge carriers are transporting the electronic heat, the thermal conductivity κe and charge conductivity σ are directly related in the diffusive regime, namely through the Wiedemann–Franz relation:The diffusion of heat is governed by the first term on the right-hand side of the thermoelectric equation, μcharge, which is the same result that can be obtained using the Einstein relation:It can be shown see ref.  that thi2 s−1, for typical EF up to 0.2 eV and mobilities varying from a few thousand cm2 V−1 s−1 for CVD graphene up to 50 000 cm2 V−1 s−1 for hBN-encapsulated graphene. Microscopic simulations predicted a diffusivity of 360 cm2 s−1.177 The diffusivity of graphene was first studied experimentally using all-optical spatiotemporal scanning,178 which found a short-lived initial  of 5500–11 000 cm2 s−1, followed by a value on the order of 250 cm2 s−1. Recently, a spatio-temporal thermoelectric scanning microscopy experiment on hBN-encapsulated graphene found a value of ∼2000 cm2 s−1.176 This value is in agreement with the measured electrical charge mobility μcharge of ∼40 000 cm2 V−1 s−1 for their sample, thus being in accordance with the Wiedemann–Franz and Einstein relations.This predicts that the electronic heat diffusivity in graphene at room temperature has a value of 100–5000 cm5.1.2ξcool, which describes how far hot charges can travel before they cool down. This is given by:179An important parameter for several hot-carrier-based applications is the cooling length EF ≫ kBTe, and owing to the Wiedemann–Franz relation, this can be written as:τmr is the momentum relaxation time that defines the electrical mobility. With typical picosecond cooling times and sub-picosecond momentum relaxation times, the cooling length is typically below 1 μm at room temperature, as observed experimentally for example in ref. et al.181 measured the cooling length using photocurrent microscopy, and found a non-monotonous trend with lattice temperature, which they ascribed to different regimes where distinctive cooling mechanisms dominated. For device configurations where optical excitation leads to a hot-carrier distribution with a spatial size larger than 1 μm, diffusion typically plays a small role.For 5.1.3Te in the heated region. This can be referred to as “diffusive cooling” or “Wiedemann–Franz cooling”, even though it is a transport phenomenon, rather than a cooling phenomenon (associated with Γcool). This cooling effect of heat transport was observed in graphene samples where carriers were heated by Joule heating, and cooling occurred by diffusion into the metal leads, cf. ref. via Johnson noise thermometry measurements, and found results that were in agreement with the Wiedemann–Franz law away from the Dirac point. These studies required cryogenic temperatures, such that other cooling and momentum relaxation channels were suppressed.Spatial diffusion of heat leads to a reduction of the average 5.1.4via the Wiedemann–Franz law. In graphene, however, it was shown that hydrodynamic transport can occur, where heat and charge flow have great similarities with the flow of a liquid in a vessel, being described by the laws of hydrodynamics. Several peculiar charge transport phenomena have been experimentally demonstrated and explained when the system is tuned to the hydrodynamic charge transport regime, such as negative local resistance.182 This occurs when the carrier–carrier scattering length is shorter than the momentum relaxation length and shorter than the device length. Hydrodynamic flow was also found to occur for phonons in graphite, in the form of the experimental observation of wave-like transport of heat, called second sound.183The above derivations – with electronic heat diffusion following Fourier's law of diffusion – hold when the scattering mechanisms affect charge and heat transport in the same way, which is usually the case. In this situation, electronic charge and heat transport are connected 99 This typically occurs very close to the Dirac point, requiring cryogenic temperatures and ultra-clean samples. However in the case of elevated Te and in the absence of electron–phonon thermalization, it can also be observed further away from the Dirac point, as demonstrated experimentally in ref.  up to 70 000 cm2 s−1 and thermal conductivities above 10 000 W m−1 K−1. This transport occurs in the hydrodynamic time window before momentum relaxation occurs (∼350 fs in their case), whereas carrier have already thermalized through carrier–carrier scattering.A special hydrodynamic charge transport regime exists very close to the Dirac point: the Dirac fluid. Here, the Wiedemann–Franz relation breaks down, and electronic heat can travel unimpeded, as electrons and holes travel together along a thermal gradient.5.25.2.1184–188 Arguably, the most prominent among them is the Seebeck effect, which refers to the conversion of a temperature difference ΔTe into an electrical voltage ΔV. According to J = 0, this conversion is proportional to the Seebeck coefficient, or thermopower, S = −∇V/∇Te. In a diffusive conductor, this effect originates from the net charge imbalance created by the thermal diffusion of carriers that have an energy-dependent DC conductivity σ(ε) = en(ε)μcharge(ε). More specifically, the relation between the electrical conductivity and the Seebeck coefficient is given by the semiclassical Mott formula:189Hot carriers in graphene can give rise to important thermoelectric and thermomagnetic phenomena.kBTe ≪ EF, which is the case for most metals and doped graphene. In practice, SMott can be calculated from the measurement of electrical conductivity of graphene at varying EF, which is conveniently achieved using a gate voltage.We note that this formula, which is based on Boltzmann theory see , is deri18,19,190–193 In these experiments, the Seebeck coefficient is typically measured using microfabricated heaters to create a temperature gradient while measuring the thermally induced voltage across the device , a voltage VPTE is created, given by:S = S2 − S1 and ΔTe is the temperature increase at the junction between the S1 and S2 regions. In the short-circuit configuration, the photovoltage acts as an electromotive force and generates a photocurrent that can be approximated as IPTE = ΔSΔTe/R, where R is the total resistance of the device. This equation is valid for a rectangular device with a uniformly heated junction. A more accurate estimation of IPTE can be obtained using the Shockley–Ramo theorem201 or a general Onsager reciprocity approach,202 which account for the geometrical factors  that govern the photoresponse that is measured between electrodes.The Seebeck effect has been extensively studied using optical excitation as a local heat source. The resulting photo-thermoelectric (PTE) effect has been shown to play an important role in the optoelectronic response of graphene devices. In this case, light absorbed in graphene generates a local distribution of hot carriers characterized by a temperature gradient ∇Te see . When th203 and graphene p–n junctions.82 By varying EF on both side of the junction, Gabor et al.82 showed that photovoltage exhibits a six-fold pattern  shown in VPTE (when S1 = S2 or EF ∼ 0 in both regions), in stark contrast with the single sign reversal expected for the photovoltaic effect. The PTE effect has also been shown to dominate the photoresponse of unbiased graphene-metal junctions114,204,205 and supported-suspended graphene interfaces.206 We note however that the photovoltaic and photo-bolometric effect can become predominant in biased graphene devices.206 In section 6.1, we discuss in more details the performance of photodetectors based on the PTE and photo-bolometric effects.The PTE effect was first observed at interfaces of single layer and bilayer graphenetern see , which i5.2.3Π = STe when an electric current is flowing through it. The Peltier heat rate generated or absorbed at the junction is given by Q̇ = (Π2 − Π1)I. This effect has been observed at graphene-metal junctions20,207 and around geometrical constrictions,208,209 typically using scanning thermal microscopy. These results, in combination with the record high thermoelectric power factor σS2 measured in clean graphene devices,196 demonstrate the advantage of using graphene as an active thermoelectric cooler. We also note that according to recent theoretical studies, the Thomson effect becomes stronger in the hydrodynamic regime199 and can lead to cooling of the Fermi liquid.210In addition to the (photo-) thermoelectric effect, several other thermoelectric and thermomagnetic processes can take place in graphene, including the Peltier, Thomson and Nernst effects. Both the Peltier and Thomson effect are included in the last term of Bz (with respect to the 2D graphene plane), hot carriers diffusing under ΔT are deflected by a Lorentz force, producing a transverse voltage Vy. This thermomagnetic effect analogous to the Hall effect is called the Nernst effect and is quantified by the Nernst coefficientSxy is the transverse component of the thermopower. The Nernst effect has been measured in graphene using microfabricated heaters and was found to agree with the generalized Mott relation, except near the charge neutrality point.18,19,191 Laser-induced electron heating was also used to probe the “photo-Nernst” effect. This effect gives rise to a photocurrent located along the free edges of the device and which exhibits a peak at the charge neutrality point.211 A large photo-Nernst current has been observed in graphene/hBN superlattices due the enhanced Nernst coefficient of the Moiré minibands, demonstrating the collection of multiple hot carriers.44Finally, in the presence of an out-of-plane magnetic field 5.3212–214 Interfacing graphene with other bulk or 2D materials has emerged as a powerful way to engineer material properties of hybrid systems. This is perhaps best exemplified by the recent discovery of anomalous superconductivity in twisted bilayer graphene.215 These heterostructures also provide the ability to control and exploit the properties of hot carriers in graphene. For instance, as discussed in section 3.2.3, encapsulating graphene using hBN gives rise to a new cooling pathway mediated by hyperbolic phonon polaritons in hBN.92 In addition to influencing the heat transfer, these heterostructures can enable the transport of carriers in the out-of-plane direction .tion see . This in5.3.1217,218 and Fowler–Nordheim's219 laws, respectively – were formulated almost a century ago to describe the emission of electrons from bulk materials into vacuum. As illustrated in ε < Φ across the barrier, which leads to a current that scales with exp(−βΦ3/2/F), where β is a constant on the material and geometry of the barrier and E the electric field across the barrier. Thermionic emission describes the process where carriers in the high-energy tail of the Fermi–Dirac distribution with ε < Φ are injected over the barrier, generating a current that scales with exp[−Φ/(kBTe)]. These early theories were later combined into a generalized model220 and further developed to describe transport in solid-state systems such as on thin films221 or III–V heterostructures.222 However, these models usually assume parabolic energy-dispersions relations for carriers in each layer.223,224The theories of thermionic and field emission – also called Richardson's223,225–228 While these models deviate from the classic Richardson's and Fowler–Nordheim laws, the aforementioned exponential behaviours hold true. These transport mechanisms have been observed in g/Si Schottky junctions229,230 and in several graphene heterostructures where layers of hBN or TMDs are employed as potential barriers.231–234 In these devices, the crystal lattices of the component layers were not aligned, so the in-plane momentum of the carriers was not conserved during transport. Indeed, the presence of disorder and phonons can lead to relaxation of the momentum conservation condition. Remarkably, by aligning the crystallographic orientation of the layers, momentum and energy conservation leads to resonant tunneling. It was experimentally shown that this can result in a strong negative differential resistance that persists up to room temperature.235,236 This coherent interlayer transport can be modelled using more advanced techniques such as ab initio simulations,224 and nonequilibrium Green's function237 and transfer Hamiltonian238 formalisms.Recently, several vertical transport models have been proposed that take into account the linear dispersion relation of graphene.5.3.2et al. predicted that hot carriers created by photon absorption could significantly enhance thermionic emission in graphene heterostructures239 and Schottky junctions.240 They showed that in some cases the heat transported by thermionic emission could even dominate other electronic cooling channels, resulting in PTI photodetectors with large responsivities. This photo-thermionic (PTI) effect was observed soon after in g/hBN/g216 and g/WSe2/g heterostructures.241 In both studies, the PTI effect is evidenced by the superlinear dependence of the photocurrent on the laser power with photon energies below the barrier bandgap , it is possible to define an interfacial Seebeck coefficient relating the voltage induced by a temperature difference across the junction. In this case, this thermoelectric voltage is due to thermionic emission of hot carriers over a thin barrier rather than bulk diffusive transport. Using electrical heating, Seebeck coefficients of ∼−220 and −70 μV K6The exceptional properties of graphene, such as its broadband absorption and ultrafast photoresponse, have led to several promising applications in photonics and optoelectronics. Indeed, graphene-based devices display many desirable features, such as high-speed operation, wide spectral response, gate tunability and compatibility with silicon photonics and CMOS platforms. Here, we focus on photonic and optoelectronic devices where hot-carrier effects play a central role, in particular photodetectors, nonlinear optical devices and light emitters.6.112,13,258–262 Graphene photodetectors can convert absorbed photons into an electrical signal through various mechanisms, including mainly the photo-thermoelectric (PTE), photo-bolometric (PB), and photovoltaic (PV) effects. While the latter is based on photoexcited carriers, PTE and PB mechanisms are driven by the excess carrier temperature ΔTe. As discussed in section 5.2.2, PTE photodetectors produce a photovoltage (or photocurrent) when hot carriers are generated at the junction between regions with different Seebeck coefficients. A hot-carrier bolometer detects incident radiation by measuring the light-induced temperature increase of the electronic bath.263 This is typically achieved by measuring changes in the temperature-dependent resistance (dR/dT) of the device or by measuring thermal noise.Over the past decade, significant efforts have been dedicated to the development of graphene-based photodetectors, as illustrated by the many review articles published on this topic.Te, which in turns generates a large PB and PTE response. Finally, carrier cooling occurs on a picosecond timescale, enabling high-speed photodetection. These unique thermal properties, combined with the broadband photon absorption of graphene, make graphene-based photodetectors highly promising for many applications. Below we concentrate on long-wavelength and ultrafast photodetectors whose performances are often on par with, or even better than, the best commercially available devices, while offering specific benefits.Both PTE and PB detectors take advantage of properties of hot carriers in graphene. As explained in section 3.1, light-induced carrier heating in graphene is efficient because the electron–phonon coupling is weak compared to the strong electron–electron interactions, possibly even leading to the generation of multiple hot carriers per incident photon. Due to the record small heat capacity of graphene, this photogenerated heat gives rise to a substantial increase in 6.1.1i.e. large bandwidth) and high sensitivity. The latter is characterized by the responsivity R, defined as the electrical output of the device per optical power input, and noise-equivalent power (NEP) of the detector, which specifies the minimum incident optical power required to achieve a unitary signal-to-noise ratio over a bandwidth of 1 Hz. Several long-wavelength photodetectors that utilize hot carriers in graphene have been demonstrated during the past decade. While these detectors directly benefit from the broadband and ultrafast photoresponse of graphene, their intrinsic sensitivity is limited by several factors, in particular the relatively low absorption of graphene.The detection of long-wavelength photons is essential for a wide variety of applications. For instance, microwave photodetectors are employed in sensitive applications ranging from radioastronomy to superconducting quantum computing, while mid- to far-infrared (IR) detectors are important for biomedical diagnostics, quality testing, thermal imaging, environmental monitoring, security and free-space optical communications. Most of these applications require long-wavelength detectors that operate at room temperature and exhibit a broad spectral response, fast photoresponse 268 limits the sensitivity of photodetectors based on the readout of the electrical resistance. One way to solve this issue is to artificially increase dR/dT, for instance, by nanostructuring graphene269 or creating a bandgap in bilayer graphene.270 Another strategy is to employ a different scheme to read out ΔTe, for example by measuring the Johnson noise of graphene104,271 or the switching current of graphene-based Josephson junctions.265One general approach to increase the sensitivity of graphene photodetectors is to optimize the photodetection mechanism which converts the excess carrier temperature ΔTe by enhancing the interaction of graphene with the incident light. This can be accomplished by integrating photonic structures, such as optical cavities, waveguides and photonic crystals, with the photodetector.272 For THz and mid-IR photodetectors, antennas with various geometries  and very fast response (∼3 ns), in particular compared with other THz detectors at room temperature.55,283 Finally, two recent studies32,265 on microwave bolometers based on superconductor–graphene–superconductor junctions reported an NEP below 1 aW Hz−0.5  dark current, resulting in high shot noise and increased power consumption.Several chip-integrated graphene-based photodetectors similar to the one shown in −1 and 12 V W−1 have been reported for PTE photodetectors. These devices typically display a bandwidth larger than 40 GHz , are also promising for high-speed photodetection as they can potentially improve the responsivity while maintaining low dark current and high bandwidth.242,294 Ultimately, improving the large-scale fabrication298 and design13 of graphene-based photodetectors could open the door to faster and more power-efficient receivers in tele- and datacom modules.In contrast, PTE detectors operate at zero-bias in either current or voltage mode, while maintaining a large photodetection bandwidth and responsivity. Like most conventional photodetectors, the current mode requires a transimpedance amplifier, whereas the voltage configuration can operate using a simpler and less costly voltage amplifier. Responsivities of up to ∼0.4 A WGHz ref.  and 292 GHz ref. .287 Fina6.2148,149 and its ease of integration in photonic devices such as fibers,153 waveguides152 and micro-resonators,300,301 graphene has been widely explored for applications in nonlinear optics. Notable examples are saturable absorbers and other optical modulators, frequency converters and sensors.15,155,259,302,303Owing to its strong and gate-tuneable nonlinear optical response6.2.1303 Both amplitude135,300 and phase304 modulation at tens of GHz have been achieved with graphene-integrated devices based on electro-optic modulation. However, the speed of electro-optic modulators is limited by the capacitance of the device.13,300 All-optical modulation overcomes this limitation and thus fully exploits the potential of graphene: the ultimate limit for the operation speed in all-optical modulators is given by the cooling dynamics of hot electrons since the absorption and/or phase modulation are defined by Pauli blocking or refractive index changes induced by photoexcited carriers. Thus, speeds well above 100 GHz are possible.Optical modulators are fundamental building blocks in a large variety of modern technological applications: phase, frequency, amplitude and polarization modulators are widely used in fibre optic communication systems, ultrafast spectroscopy, metrology, active Q-switching or mode-locking of lasers and quantum information. The scope of integrated optical modulators is to encode information into light with faster response time and lower power consumption compared to electrical interconnects. Graphene can fulfill all these requirements and, in addition, can be easily integrated with standard photonic platforms such as waveguides, microresonators and fibres.155,303 Graphene SAs have been used to mode-lock both solid-state305–307 and fibre lasers154,308–312 at 800 nm,307 1 μm,311 1.5 μm,154,308 2 μm (ref. 313 Saturable absorption based on σintra (see section 4) is possible also at THz frequencies.173,314 Graphene SAs have enabled the generation of ultrashort pulses with <70 fs duration307 and lasers with GHz repetition rate.301,310 Several other approaches have been implemented for the realization of graphene-based all-optical amplitude modulators. With a graphene-clad microfiber , a nonlinear and non-parametric third-order process that has been extensively used for passive mode-locking of ultrafast lasers .155,303  μm ref.  and 311 crofiber  Li et al6.2.2.148 and FWM149,152 (see section 4.1.3). Recently, ultrafast all-optical modulation with a response time of ∼2.5 ps, thus limited by the hot-electron cooling time, has also been observed.319 Frequency conversion in the THz range is another application which, through the thermodynamic nonlinearity mechanism, is fully enabled by hot electrons. The attractiveness of graphene in this field comes from its small footprint and high efficiency, as shown by observation of the record value of 1% conversion efficiency obtained with a grating-graphene metamaterial using a moderate field strength of ∼30 kV cm−1 ,320 enabling its incandescence via Joule heating. Early studies25,78 showed that the emission spectra of electrically biased graphene devices agree well with Planck's law for the spectral radiance  of a grey body,ħω is the photon energy, c is the speed of light and Ξ is the emissivity of the grey body. An emissivity value of ∼2.3% was measured,25,78 in agreement with Kirchhoff's law stating that the emissivity and absorption are equal. Electronic temperatures Te on the order of 1500 K were measured using this technique, corresponding to a peak emission in the infrared (see −6) was attributed to the fact that Joule heat is mainly dissipated into the metallic contacts and through the SiO2 substrate via surface polar phonon scattering.In addition to detecting and modulating light, graphene can also serve as a broadband and ultrafast thermal light emitter. The superior mechanical, thermal and electronic properties of graphene allow it to sustain very high current densities  and brighter emission. However, suspended devices must operate in vacuum since graphene quickly oxidizes at high temperature. hBN encapsulation was found to provide excellent protection for graphene even at temperatures above 2000 K.320–324 These emitters have a measured efficiency of ∼1.6 × 10−5 and an estimated lifetime exceeding 4 years.320,321 These results were attributed in part to the efficient cooling of the hot carriers into the hBN mediated by coupling to hybrid polaritonic modes (see section 3.2.3). Finally, another avenue to enhance Joule heating and thermal radiation in graphene is to confine the current flow through a narrow constriction.324Several approaches have been investigated to improve the emission efficiency of graphene emitters. To reduce vertical heat dissipation, Kim ters see  and demo26,320,321,324,325 and photonic crystals.323 The latter enabled the strongest modulation of the black-body radiation spectrum in the near-infrared region, which is relevant for optical communications.Furthermore, the broad spectral emission of graphene emitters can be easily tailored to meet the needs of specific applications. This can be achieved by integrating the graphene emitter in a sub-wavelength optical structure which modifies the electromagnetic local density of state. Different structures have been demonstrated, including photonic microcavities28,320,323 reported a temporal response of ∼100 ps , where the electron–phonon coupling is drastically reduced.328,332 Alternatively, hot carriers can also be generated by short intense laser pulse. In this case, Te is determined by the heat capacity coefficient γ  is roughly two orders of magnitude larger than that of graphene 331,334 is significantly longer than in graphene (<100 fs),37,45 which leads to higher heating efficiency in the latter. All these differences help explain the preponderance of hot carrier effects in graphene as compared to conventional metals.As we have shown throughout this review, hot carriers play an important role in the electrical and optical response of graphene. We note that semiconducting two-dimensional material systems can also exhibit important physical effects related to hot carriers. Nevertheless, the vanishing band gap, linear energy–momentum dispersion relation, and strong carrier–carrier interactions of the Dirac fermions in graphene set graphene apart. One important consequence of these properties is graphene's small electronic heat capacity ions see  and for nt γ see , which i7.22Se3 338 showed the occurrence of negative photoconductivity (a decrease in THz conductivity after photoexcitation). This is similar to OPTP experiments on graphene, where negative photoconductivity was interpreted as a signature of hot carriers.34 Time-resolved ARPES measurements with visible pump light performed on Bi2Se3 339,340 and Bi2Te3 341 discussed the photo-induced carrier thermalization and cooling dynamics. In contrast with graphene, in these topological-insulator systems, the presence of bulk bands leads to complex dynamics involving an interplay between bulk and surface charges. There have been several attempts to eliminate the effects of bulk charges, for example by studying topological insulators excited by THz and mid-infrared light, such that no interband absorption takes place. For example, Luo et al. recently used this approach to demonstrate faster dynamics for hot surface-state carriers compared to hot bulk carriers, for Bi2Se3 at 5 K.324 At room temperature, the experimentally observed dynamics became similar, which is due to efficient surface–bulk coupling, as explained in ref. Besides graphene, there are several other material systems with Dirac electrons, characterized by a linear energy–momentum dispersion relation. It can be expected that similar hot-carrier effects as in graphene play a role in these materials. We will discuss several quantum materials, starting with topological insulators. These are material systems that are semiconducting in the bulk, whereas on the surface (for 3D systems) or edge (for 2D systems), protected conducting states emerge with nearly linear dispersion . Not surprisingly, it has been shown that significant photo-thermoelectric currents can be generated,346,347 similar to graphene. Furthermore, using the 3D semimetal Cd3As2, ultrafast, broadband photodetection based on the photo-thermoelectric effect has been demonstrated.348 Also in the field of nonlinear light conversion, different Dirac materials have generated interesting results. Terahertz nonlinearities, for example, have been shown to originate from Dirac surface states in Bi2Se3,349 and, very recently, THz harmonics have been generated from Cd3As2.163,350 We should note that in the latter case, the role of the hot carriers is debated, as these results were described using coherent carrier acceleration, rather than the thermodynamic nonlinearity that is responsible for the exceptionally large THz nonlinearity in the case of graphene . Besides continuing the pursuit of a detailed understanding of the fundamental physics of hot carriers in graphene itself, a highly promising research direction is that of graphene interfaced with other (2D) materials. We have discussed a few examples of such systems, where the substrate of graphene, or adjacent 2D materials, play an important role in determining the overall system properties, in particular related to transport (see section 5). Another highly interesting avenue is that of graphene interfaced with graphene itself. For example, we note that exciting hot-carrier dynamics have been observed for 30°-twisted bilayer graphene,There are no conflicts to declare.


Simulations and experimental achievements carried out during more than two years of investigations are reported here, detailing the technical expedients that were necessary to increase the single layer CVD graphene absorption first to 39% and then up to 84%. Graphene absorption increased when an asymmetric Fabry–Perot filter was applied rather than a symmetric one, and a further absorption increase was obtained when graphene was embedded in a reflective rather than a transmissive Fabry–Perot filter. Moreover, the effect of the incident angle of the electromagnetic radiation and of the polarization of the light was investigated in the case of the optimized reflective Fabry–Perot filter. Experimental challenges and precautions to avoid evaporation or sputtering induced damage on the graphene layers are described as well, disclosing some experimental procedures that may help other researchers to embed graphene inside PVD grown materials with minimal alterations.Graphene is emerging as a promising material for the integration in the most common Si platform, capable to convey some of its unique properties to fabricate novel photonic and optoelectronic devices. For many real functions and devices however, graphene absorption is too low and must be enhanced. Among strategies, the use of an optical resonant cavity was recently proposed, and graphene absorption enhancement was demonstrated, both, by theoretical and experimental studies. This paper summarizes our recent progress in graphene absorption enhancement by means of Si/SiO Graphene-based absorbers are receiving a considerable interest due to their potential applications in photovoltaics ,2,3, as In the THz and IR regions, however, high quality graphene may show strong interaction with light thanks to generation of surface plasmon polaritons (SPPs), making graphene a promising alternative to typical plasmonic materials ,23,24,25In addition, tunable graphene electromagnetic response may be achieved by chemical and/or electrical doping or by using a magnetic field ,31,32.In the Vis and NIR regions, on the contrary, absence of SSPs makes it necessary to couple graphene with resonant structures such as metamaterials, photonic crystals and plasmonic materials ,37,38,39Many examples of perfect graphene-based absorbers have been reported in the literature basing on the critical coupling concept ,41,42.Thongrattanasiri et al. have simulated 100% absorption in doped graphene nanodisks by exploiting the critical coupling conditions .Piper et al.  have demLiu et al.  have demXiao et al.  have repJin et al.  have revAnother strategy to enhance graphene absorption is to exploit the electric field enhancement by resonant cavities ,49,50. TNulli et al. reported theoretically that the optical absorption of single-layer graphene can be enhanced up to 50% by increasing the electric field on the surface of undoped graphene by placing graphene inside symmetric Fabry–Perot structures .Ferreira et al. have calculated the same absorption result for single layer graphene by utilizing one Fabry–Perot cavity and they have simulated also that graphene optical absorption can increase up to 100% using two symmetric Fabry–Perot cavities .Zand et al. have used a genetic optimization algorithm coupled to a transfer matrix code to design one-dimensional aperiodic multilayer microstructure embedding single layer graphene, where near-total absorptions at selected wavelengths is obtained by existence of critical coupling . They siFurchi et al.  have dem2/Vs, which is generally much higher than the real one , leading sometimes to too overoptimistic predictions [Many of the theoretical studies found in the literature simulating perfect graphene absorption consider for single layer graphene a very high mobility µ of about 10,000 cmdictions ,65.Apart from the work from Furchi et al., the enhancement of graphene absorption by using an optical resonant cavity has been rarely explored experimentally due to the many experimental challenges related to embedding graphene inside metals or dielectric stacks ,66.The graphene absorption enhancement inside an optical resonant cavity is generally narrowband and is, therefore, more suitable for application in photodetectors, sensors, or absorption filters ,66,67.In the present work, we summarize our recent studies ,68,69 onIn the optimized structure, a high absorption of 84% at 3150 nm was measured  in case 2 (L) layers, for use in the near to mid IR. Three different Fabry–Perot filters were simulated: (i) a symmetric FP filter with identical top and bottom mirrors (FP1), (ii) an asymmetric FP filter where the bottom mirror had a higher reflectance due to a larger number of layers with respect to the top mirror (FP2), and (iii) an asymmetric reflective FP filter obtained by further increasing the bottom mirror layers (FP3). FP1 and FP2 filters worked in transmissive mode, while FP3 worked in reflective mode, due to the high reflectance of the bottom mirror (99.7%). The structure of FP1, FP2, and FP3 filters was the following: (i) air/HLH LL HLH/sub, (ii) air/HLHL HH LHLHLH/sub, and (iii) air/HLH LL HLHLHLHLH/sub, respectively. For FP1 and FP3 the cavity was (LL), i.e., constituted by a half wavelength thick SiO2 layer, for FP2 the cavity (HH) was made by a half wavelength thick Si layer. This last FP structure, a bit different from the other two, was chosen because of its particular distribution of the electric field characterized by two maxima. A single layer graphene (SLG) was embedded inside the FP structure and located at the position where the electric field was maximum to enhance its absorption. SLG was covered by a 30 nm thick MgF2 layer, meant to protect graphene during the later sputtering deposition. A Fabry–Perot optical filter consists of a cavity separated by two flat and parallel high reflecting mirrors where light experiences multiple reflections. Simulation of FP filters was carried out by using the Wave Optics Module of Comsol Multiphysics software. Our Fabry–Perot filters consisted of a sequence of alternate quarter wave thick high refractive index Si (H) and low refractive index SiOG) where SLG is positioned. The central wavelength of the Fabry–Perot filters is different for the three cases because it results from different experiments, however, central wavelength position in the considered wavelength range, only causes small variations of T, R, and A due to the wavelength dependence of Si and SiO2 optical constants. As we can observe from 2 with optical thickness equal to a quarter wavelength . Si and SiO2 layers were fabricated by radio frequency sputtering in MRC  systems, starting from 99.999% purity Si and/or SiO2 targets, as described elsewhere [2 layer covering graphene, a radiofrequency sputtering of 200 W and a sputtering pressure of 1 Pa were used. For the first 30 nm of SiO2 layer deposited onto graphene, a milder sputtering condition was used: i.e., sputtering power of 50 W and sputtering pressure of 3 Pa. For FP3 filter, grown in a different system, the same sputtering conditions (sputtering power of 200 W and sputtering pressure of 0.53 Pa) were used for Si and all the SiO2 layers, due to the fact that such system geometry allowed a milder sputtering process.The Fabry–Perot filters consisted of alternate layers of Si and SiOlsewhere ,68,69. F2 layer was carefully evaporated in a Balzer BAE 250 evaporation system, and the evaporation rate was controlled by a quartz crystal microbalance.MgF2. After the growth, graphene was removed from the back of the Cu substrate by means of an oxygen plasma cleaner . For graphene transfer, a procedure reported elsewhere was applied [Single layer graphene was grown by CVD on Cu foils previously submitted to a pre-oxidation treatment at 250 °C for 90 min. CVD growth conditions were the following: substrate temperature 1070 °C, 10 mbar pressure by using 0.025 SCCM ethanol vapor with 20 SCCM Ar and 10 SCCM H2 layers which separate the top and bottom mirrors. In the asymmetric FP2 filter, two maxima of electric field occurred, and graphene was positioned onto the upper Si layer of the cavity (HH). In the asymmetric reflective FP3 filter, graphene was positioned in the middle of the cavity (LL), i.e., sandwiched between the two SiO2 layers.2 protective layer. After the evaporation of the MgF2 layer, the two samples were transferred back into the sputtering chamber for the deposition of the top multilayer structure.The fabrication procedure of the multilayer structure was the following. First, the bottom multilayer structure and two CVD graphene layers were fabricated separately in the sputtering and CVD chambers, respectively. For each sputtering deposition run, two substrates (quartz or Si) were loaded in the chamber, one for the reference and the other for the graphene-based FP filter. Similarly, two graphene layers were prepared in the same run, one for insertion into the FP filter and the other for characterizations. Then, the graphene layer was transferred on the top of the bottom multilayer by the mild transfer process described above. The multilayer structure topped by graphene and the reference multilayer (without graphene) were transferred into the evaporation chamber for the growth of 30 nm MgF−1 in the latter case. For FTIR measurements, a Fixed-Angle Specular Reflectance Accessory from Perkin Elmer was used.Transmittance and reflectance measurements in the NIR were carried out with a PerkinElmer 900 spectrophotometer, while MIR measurements were performed with a high resolution FTIR Perkin Elmer instrument. The reflectance measurements were obtained by using calibrated standards, i.e., an Ocean Optics Al mirror certified in the range 250–2500 nm in the first case and an Al infrared reflectance standard with a SiO overcoat certified by the National Physics Laboratory (NPL) in the wavenumber range of 4000–200 cmThe Raman measurements were carried with a Renishaw inVia Reflex Raman spectrometer using a 514.5 nm excitation source.tG (0.35 nm) [σ(ω) the intrinsic contribution to the graphene relative permittivity, the thickness of single layer graphene and the graphene optical conductivity, respectively.ω)=εg(ω) ,74, wherω)=εg(ω) ,76,77:εg εb(2.5) ,79, tG (0.35 nm) ,81, and The graphene optical conductivity was considered as the sum of interband transitions and Drude-like intraband conductivity ,83,84,85= (e24ℏ) ,86, Ef tlsewhere .Single layer graphene permittivity was simulated also as a function of the wavelength and Fermi level, as shown in 2 evaporation, after SiO2 sputtering on MgF2/graphene layers and after the whole filter fabrication process.The quality of graphene was checked after each experimental step, i.e., after MgF2 layer directly onto SLG, a mild sputtering condition was applied by increasing the sputtering pressure and decreasing the radiofrequency power. Notwithstanding, graphene damage could not be avoided, as demonstrated by Raman analysis (not shown here), and a further graphene protection was necessary. Therefore, to avoid the sputtering of the SiO2 layer directly onto graphene, a thin protective MgF2 layer was evaporated prior to the sputtering of SiO2. We have chosen MgF2 due to its refractive index and extinction coefficient values, very close to those of SiO2. This is necessary since MgF2 becomes part of L (the low index layer) of the multilayer stack. Other fluoride materials could have served the purpose. Furthermore, the evaporation of the MgF2 layer required some care since we observed that if evaporation occurred at a high rate, graphene defects band D increased. 2 at high (2 Å/s) and low (0.1 Å/s) evaporation rates. Only the latter case left graphene unchanged. The effect of deposition rate of SiO2 was preliminarily evaluated in experiments with ML graphene with unoptimized MgF2 deposition (curve of 2 deposition on SL graphene can be seen in the red and blue curves of 2), suggests that some extent of the modifications of the D and 2D bands by SiO2 is not due to lattice damage but to optical and electronic “proximity” influences on graphene by the stack.Initially, for the growth of the first 30 nm of SiOcurve of b. The ef2 and SiO2 deposition  evaporation of the thin MgF2 layer should occur with a very low rate, (iii) the presence of the thin MgF2 layer is not by itself sufficient to avoid the damage if the sputtering process is too energetic, (iv) the combination of evaporated MgF2 and mild SiO2 sputtering increases the graphene D band but preserves the optical and electrical properties of pristine graphene, finally (v) we cannot certainly assign the measured D band increase to an effective graphene “damage” while inside the stack due to possible proximity effects by the embedding layers on the Raman spectrum itself.The results obtained after several attempts to not damage graphene can be summarized as following: (i) a thin MgFR is the reflectance of the top and bottom mirrors, while on the right-hand is the finesse of the asymmetric filters (FP2 and FP3). In our FP filters, the finesse increased from 21 to 31.5 when passing from the symmetric structure of Above, the left-hand side formula represents the finesse of the symmetric FP filter (FP1), where Absorption of unpatterned and undoped single layer graphene up to 84% has been never reported experimentally in a Fabry–Perot filter. Comparable or even higher absorption values have instead been obtained by other authors by using graphene coupled to non-planar resonant plasmonic structures ,91,92.Enhancing the light-matter interaction in single-layer graphene by the use of optical microcavity requires no stringent constrains on the graphene electrical properties, such as the sheet resistance, and may find application in a variety of graphene-based devices, such as optical absorption modulators, light emitters, and optical attenuators, and provides new routes to graphene photonics for applications in spectroscopy, communications, sensing, and security.Results, validated here at NIR and MIR wavelengths, may be applied in principle at different wavelengths ranging from Vis to THz by properly choosing the materials of the layers and of the substrate. Moreover, as obtained by our previous simulations, modification of the filter amplitude and bandwidth may be achieved by: (i) using multilayer graphene instead of single layer graphene , (ii) usResults are in accordance with what has been reported by other authors. In , Xiao etGraphene absorption discussed so far was obtained in the case of a TE polarization. For the optimized FP structure, i.e., the FP3 filter, the simulation was carried out, also, in case of TM polarization. 2 layers. Results demonstrated that graphene absorption greatly increases when graphene is embedded inside an asymmetric Fabry–Perot structure, reaching its maximum in case of a reflective Fabry–Perot filter. SLG properties were preserved during the sputtering process by applying a thin, slowly evaporated MgF2 layer, allowing the effective embedding of graphene inside thick monolithic structures fabricated by conventional PVD techniques. Such a high graphene absorption discloses exciting potentiality for exploitation of 2D materials in new optoelectronic devices for application in the NIR-MIR spectral range.This work summarizes our recent findings on single layer graphene absorption when embedded in dielectric multilayer stacks. The increase of single layer graphene absorption obtained by exploiting the electric field enhancement inside a resonant optical cavity was modeled and experimentally demonstrated in three different Fabry–Perot filters with central wavelengths varying in the NIR-MIR spectral ranges. The Fabry–Perot filters were fabricated by radiofrequency sputtering, and consisted of alternate quarter wave thick Si and SiO


We performed ab initio numerical simulations with the density functional theory to investigate the variations in the band structure, optical absorption, and the reflectivity of vacancy-graphene doped with nitrogen, oxygen, and fluorine for different densities. We considered the density values 0.78%, 1.02%, 1.39%, 2.00%, 3.12%, 5.55%, and 12.5% for the vacancies and doping. In the infrared and visible ranges for all cases, vacancies included, there is a substantial increment in the absorption and reflectivity concerning graphene. The most significant changes are for fluorine and oxygen at a concentration of 12.5%. The fact that semiconductors have electrical and optical properties tunable by changes in the band structure is one of continuing importance and interest. Recent research on graphene and other 2D materials  has focuOne of the main ways of tuning the graphene’s bandstructure is doping it using different atoms. Using nitrogen has shown to be relevant for energy, sensor development, and environmental applications ,6. OxygeCombining nitrogen and fluorine to dope graphene resulted in good electrochemical performances when tested as the anode for lithium-ion batteries . HoweverOther authors’ studies focused on tuning the band structure via the symmetry of the dopant atoms to the graphene surface and modifying the graphene positions concerning the substrate ,20. StudThis work explores how graphene’s band structure and its optical properties change by varying graphene vacancy density or dopant concentration. The band structure, optical absorption, and reflectivity variations are useful tools for electronic and optical applications and detecting dopants and defects in the surface. We considered fluorine, nitrogen, and oxygen as dopants in the graphene vacancies. The concentration values we took were 0.78%, 1.02%, 1.39%, 2.00%, 3.12%, 5.55%, and 12.5%. We performed first-principles numerical simulations.We considered seven configurations in terms of the dopant density, performing structural relaxations for each. The graphene surface’s most prominent deformation occurs for the F dopant–this case is as F-graphene. The N dopant–labeled as N-graphene–incorporates smoothly into the lattice without deforming it, apart from a slight displacement of 0.1 Å over the graphene plane, observed only at dopant densities of 5.55% and 12.5%. For the case of O-graphene, while increasing the dopant density, its distance from the surface slightly decreased, remaining approximately 0.5 Å from the graphene sheet.Adding dopants of similar size to carbon could be incorporated into the lattice while preserving the structure, as shown when nitrogen and oxygen adsorbed in the graphene’s plane. Whereas fluorine, being more prominent and with one missing electron to fill its shell, tends to bond with only one carbon. Furthermore, this bond deforms the graphene lattice, creating a bulge in the system.All oxygen and nitrogen percentages bounded with the three-carbon surrounding the vacancy, with an average bound size of 1.53 and 1.42 Å, respectively. O-graphene has a bond length of 0.11 Å greater than pristine graphene, bringing oxygen slightly out of the graphene’s plane. In comparison, F-graphene is bound to one carbon, with an average bound distance of 1.38 Å.The distances between carbons remain almost constant regardless of the dopant percentages. The separation between the first neighbors is an average of 1.3871 Å in the vacancies cases. F-graphene maintains an average distance of 1.4021 Å. O-graphene keeps an average spacing of 1.3889 Å (same as with vacancy), while in the case of nitrogen, the average spacing is 1.4151 Å.s and p states’ hybridization for the lowest vacancy density is mainly in the Fermi energy neighborhood. As this density increases, the s and p hybridization spreads beyond the Fermi energy.In s and p around the Fermi energy for carbon in all cases. We can observe a general tendency. The orbitals s from carbon and oxygen hybridize with the orbitals p from oxygen and carbon in tiny contributions below the Fermi energy. As the dopant density increases, this hybridization occurs at lower points, and the contributions from orbitals s grow. We must mention that this general tendency fails for the densities of 1.39% , this hybridization occurs above the Fermi energy at 2.9 eV.s orbitals are meager. In all cases, the hybridization occurs at the Fermi energy. We can observe the hybridization of orbitals s and p above the Fermi energy in all cases. The hybridization is more intense as the density increases. We can see the same kind of hybridization below −0.5 eV.In In The surfaces showed some relevant band gap variations. The first column on the left corresponds to pristine graphene for seven different unit cells. Each cell contains 2N × N atoms. The 12.5% of dopant (or vacancy) corresponds to a unit cell of 8 bits (N = 2). 5.55% compares to a unit cell with N = 3; 3.12% refers to a unit cell with N = 4; 2.0% corresponds to 50 atoms (N = 5). 1.39% refers to N = 6; 1.02% refers to a unit cell with N = 7 (98 bits); 1.39% compares to a unit cell with N = 8. Finally, 0.78% corresponds to a unit cell with N = 8 and 128 atoms.We can observe that the Dirac cones appear at the Κ points, except for the unit cells that contain 18 and 72 atoms . These unit cells follow the 3N × 3N rule . In thesWe may say that, in general, the structures are semimetallic. For the vacancies, there is a quasi-flat energy band, around the Fermi energy, for concentrations below 2%. Flat energy bands could imply many-body phenomena . There is another maximum for a concentration of 12.5% that occurs at 3.2 eV .fluorine as the dopant, again, the optical absorption increases quickly with concentration. In general, the absorption increases with all concentration values’ energy radiation. However, the maximum absorption occurs for a concentration of 12.5% at the energy around 1.6 eV . After that, the absorption decreases with increasing radiation energy. The maximum absorption is closely 1500 times the absorption of graphene.With nitrogen as the dopant, the optical absorption magnitude is two or three times larger than the graphene’s corresponding absorption. The two significant peaks happen for concentrations of 5.55% and 12.5%. The first one occurs around 2.75 eV , and the second at about 3 eV .In oxygen is the dopant, for concentrations below 2%, there is a peak value for the optical absorption. This peak occurs at 2.25 eV . Furthermore, in general, the absorption increase as the dopant concentration increments. When the dopant percentage is 12.5%, the absorption grows with frequency. In this case, the absorption is around one thousand times the corresponding absorption of graphene.When pristine graphene shows two prominent peaks. One is for incident radiation of 11.02 eV, and the other for 14.58 eV. The absorption resembles pristine graphene for vacancy concentrations below 5.55%; when the percentages are 5.55% and 12.5%, the maximum on the left moves about 1.5 eV to the right.In fluorine is the dopant, the absorption, again, resembles pristine graphene for dopant percentages below 5.55%, for concentrations 5.55% and 12.5%, the maximum on the left moves to the right about 2.5 eV. In this way, the two maxima come close to each other. Furthermore, for 12.5% of dopant concentration, three small peaks appear below 7.5 eV.When nitrogen case, the changes are tiny. The most noticeable is 12.5% concentration, where the peak on the left moves about 1 eV to the right.In the oxygen is the dopant, there are few changes concerning pristine graphene for all dopant concentrations, except for 12.5%. There is a tendency to move the peak on the right slightly to the left as the percentage increases. However, for 12.5%, the maximum on the right moves about 1.5 eV to the left. Furthermore, the peak on the left moves about 5.0 eV to lower energies.When pristine graphene shows two maxima located in the ultraviolet range. One is for radiation of 11.07 eV, and the other for 14.57 eV. For vacancies, we see that for all concentrations, the reflectivity resembles pristine graphene. There is only a noticeable change at 12.5%; the peak on the left moves 0.3 eV to the right, getting closer to the other maximum.In fluorine is the dopant for all concentrations, the reflectivity resembles graphene. There is a significant change for 12.5%. In this case, the peak on the right moves 0.1 eV to the right. Furthermore, the maximum on the left disappears, and five peaks appear below 8 eV.When nitrogen, the reflectivity is similar to that of graphene for all dopant concentrations. There are tiny changes only when the percentage is 12.5%. In this case, the two peaks move to the right about the same amount, 0.12 eV.In the case of oxygen as a dopant happens something similar to fluorine. In general, the reflectivity resembles graphene for all concentrations except at 5.55% and 12.5%. In the case of 5.55%, one additional maximum appears between the two prominent peaks. At 12.5%, the maximum on the right moves 0.15 eV to the left. The peak on the left disappears, and three maxima appear below 8 eV.Finally, with We investigated the changes in the structure of the energy bands and the optical properties of graphene when the concentration of vacancies and doping changed. We considered fluorine, nitrogen, and oxygen as dopants. We investigated the concentration values 0.78%, 1.02%, 1.39%, 2.00%, 3.12%, 5.55%, and 12.5%.The three dopant atoms chemisorb in the graphene surface, with absorption energies that suggest stability of the resulting systems. In the three cases, the magnitude of the adsorption energies decreases with increasing doping density almost linearly. We found that fluorine showed the lowest adsorption energy and nitrogen the highest. Nonetheless, we found that the dopant density influences the hybridization of the atomic orbitals and the energy band structure. Oxygen as a dopant at a concentration of 12.5% presents the largest bandgap (0.95 eV). Fluorine shows a maximum bandgap of 0.8 eV at a concentration of 3.12%. The widest bandgap we obtained for nitrogen as a dopant was 0.70 eV at a concentration of 12.5%.We found substantial variations in the optical properties when the vacancy and doping concentrations change. The most significant changes in reflectivity are with fluorine and oxygen as dopants. Five maxima appear below 8 eV, where the pristine graphene shows a near-zero reflectivity in fluorine at a concentration of 12.5%. The highest of these maxima has an intensity of 20% of the maximum reflectivity in pristine graphene. One of the two prominent peaks in graphene disappears. Something similar happens with oxygen: three peaks appear below 8 eV at 12.5%. The most intense of these peaks is 22% of the highest maximum in pristine graphene. One of the two prominent peaks in pristine graphene also disappears.For the optical absorption, there are only two maxima in pristine graphene and occur in the ultraviolet range. In the rest of the spectrum, the absorption is near zero. In the infrared, in all cases (including vacancies), we obtained significant changes concerning pristine graphene. Fluorine, at a concentration of 12.5%, shows the most significant change. The most prominent peak was 4.0% of the maximum absorption in pristine graphene. We should mention that the second-largest absorption occurred for the vacancies and was 3.6% of the most prominent peak in graphene.In the visible range, something similar happens. For all cases, vacancies included, there is a substantial increment in the absorption. The most significant changes are for fluorine and oxygen as dopants at a concentration of 12.5%. The maximum absorption for fluorine was 13.6% of the most considerable absorption in graphene and for oxygen was 8.2%.For the ultraviolet region, the most relevant changes concerning pristine graphene occur at a concentration of 12.5% for all instances, including vacancies. However, for oxygen and fluorine, the peak on the left disappears. In oxygen, three maxima appear below 8 eV. The maximum of these three is 27.2% of the maximum absorption in graphene. In fluorine, five maxima appear below 8 eV, and the most prominent of these reaches 22.7% of the maximum peak for pristine graphene.We found significant variations in the optical properties, especially in the infrared and visible ranges, implying changes in the material’s color. The changes in the tincture of defected or doped graphene may be a valuable tool for detecting the concentration of dopants or vacancies.−6 eV. We considered a supercell size large enough to avoid spurious interactions. In this way, the cell size on the z-axis was 15 Å.We utilized density functional theory (DFT) via the Quantum ESPRESSO code ,29 to pX = N, O, F.Many studies are focused on calculating the properties when the dopant is inside the graphene structure. We perform a geometrical optimization with the dopant out of graphene’s plane to analyze which kind of interaction would be natural. Thus, we placed each dopant atom at 3 Å from the vacancy graphene sheet. In all cases, we optimized the graphene surface with vacancies. We then added the dopant and performed a geometry optimization. We calculated the adsorption energies using the formula (1):For visualizations, we used the XCrySDen software . We calcn is the electron density, m is the effective mass, and ω is the photon frequency. The indices α and β are the cartesian components of p that is the vector defining the incident electric field’s polarization. The symbols ck and kυ are the wave functions corresponding to the conduction, and the valence bands with crystal wave vector k; fc is the Fermi distribution function for the state c. The sum of the transitions from occupied to unoccupied states is over the first Brillouin zone. Finally, we weigh the probability of a change.In Equation (2), By the Kramers-Kronig relations  and usinRii and Aii are :Considering a perpendicular incidence of the electromagnetic wave, the reflectivity and absorption And