Nanoscale View Article Online REVIEW View Journal | View Issue Hot carriers in graphene – fundamentals and This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. Cite this: Nanoscale, 2021, 13, 8376 applications Open Access Article. Published on 29 April 2021. Downloaded on 12/5/2022 12:11:46 PM. Mathieu Massicotte, a Giancarlo Soavi, b,c Alessandro Principid and Klaas-Jan Tielrooij *e 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 par- ticular interest are optical and optoelectronic applications, where hot carriers are used to detect ( photo- detection), convert (nonlinear photonics), or emit (luminescence) light. Graphene-enabled systems in Received 28th December 2020, these application areas could find widespread use and have a disruptive impact, for example in the field of Accepted 30th March 2021 data communication, high-frequency electronics, and industrial quality control. The aim of this review is DOI: 10.1039/d0nr09166a to provide an overview of the most relevant physics and working principles that are relevant for appli- rsc.li/nanoscale cations exploiting hot carriers in graphene. 1 Introduction Understanding and controlling the properties of hot carriers – a Institut Quantique and Département de Physique, Université de Sherbrooke, electrons and holes with excess kinetic energy – is a paradig- Sherbrooke, Québec, Canada matic topic in both physics and chemistry. In graphene, hot b Institute of Solid State Physics, Friedrich Schiller University Jena, 07743 Jena, carriers are particularly consequential because they can be Germany efficiently created, controlled and exploited towards appli- c Abbe Center of Photonics, Friedrich Schiller University Jena, 07745 Jena, Germany d cations, as we will discuss in detail in this review. Hot carriers School of Physics and Astronomy, University of Manchester, UK e Catalan Institute of Nanoscience and Nanotechnology (ICN2), BIST & CSIC, in graphene have been addressed experimentally1–4 and expli- Campus UAB, 08193 Bellaterra, Barcelona, Spain. E-mail: klaas.tielrooij@icn2.cat citly through theory,5,6 since around 2007, although they have Dr Mathieu Massicotte is a post- Giancarlo Soavi obtained a PhD doctoral research fellow at the in Physics from Politecnico di Institut Quantique of Université Milano in 2015 and sub- de Sherbrooke in Canada. He sequently worked as a Research completed his PhD in Photonics Associate at the Cambridge in 2017 at ICFO – The Institute Graphene Centre. From 2019 he of Photonic Sciences in Spain is a tenure track Professor at the after obtaining his master’s Friedrich Schiller University of degree in Physics from McGill Jena. His research interests focus University. His research interests on ultrafast spectroscopy and lie at the intersection of experi- nonlinear optics in quantum con- mental condensed matter physics fined systems, including gra- Mathieu Massicotte and photonics, with a focus on Giancarlo Soavi phene and related layered optoelectronic nanodevices materials. based on novel 2D materials. 8376 | Nanoscale, 2021, 13, 8376–8411 This journal is © The Royal Society of Chemistry 2021 View Article Online Nanoscale Review been the subject of intense research in graphene-related materials, such as graphite and carbon nanotubes, since many years cf. ref. 7–10. Now, more than a decade after the first studies, the understanding of hot-carrier phenomena in gra- phene and graphene-based hybrid systems has led to the realization that their properties enable highly promising appli- cations. These insights have already led to a significant number of patents and business creations, besides a large number of scientific publications. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. In this review we will focus on the aspects of hot carriers that are most relevant for applications involving light, and on Open Access Article. Published on 29 April 2021. Downloaded on 12/5/2022 12:11:46 PM. systems where graphene is the active material containing hot carriers. Our aim is not to provide a fully exhaustive review of Fig. 1 Illustration of heating–cooling dynamics of graphene carriers, all relevant literature in this expansive and expanding field. where in the hot state the carrier distribution is characterized by an We refer to several reviews with relevant background infor- increased carrier temperature Te that is larger than the lattice tempera- mation: ref. 11 provides an excellent introduction to hot car- ture TL and a “hot” chemical potential μ that is smaller than the equili- brium chemical potential μeq. riers, with a focus on metallic nanostructures. Ref. 12 and 13 discuss photodetection and data communication applications of graphene (and related 2D materials), respectively. In ref. 14 and 15, nonlinear optical effects and applications of 2D As a starting point for describing the relevant physics of hot materials are reviewed, and ref. 16 and 17 provide reviews on carriers in graphene, we provide the thermoelectric equation energy dissipation in graphene (and related 2D materials). describing the time evolution of the electronic temperature Te, We distinguish two types of hot carriers, where in both assuming that the electron density remains constant over cases charge carriers have an increased kinetic energy. We will time t: use the term “high-energy charge carriers” for those that have an increased energy, yet do not form part of a thermalized @Te Ce ¼ Q̇ext þ κe ∇ 2 Te  ∇  ½ðV þ ΠÞJ ; ð1Þ carrier distribution. In this case, it is not possible to define a @t |{z} |fflfflfflffl{zfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} heating=cooling diffusion thermoelectrics meaningful temperature, even if the carrier energy can be expressed in units of temperature. We will exclusively use the where Ce is the electronic heat capacity. The first term on the term “hot carriers” for those high-energy carriers that have right, Q̇ext, is the rate at which heat is added by external thermalized with other charge carriers, thus forming a Fermi– sources and removed via internal or external cooling channels, Dirac distribution with an increased temperature – a hot- for example via electron–phonon collisions. The other two carrier distribution (see Fig. 1). In this case, the temperature of terms on the right-hand side are related to transport phenom- the electronic system Te is higher than the (equilibrium) temp- ena. The second term corresponds to Fourier’s law of heat erature of the lattice TL. conduction, and contains the electronic part of the thermal Dr Principi is Lecturer of the Klaas-Jan Tielrooij obtained his Department of Physics and Ph.D. from the University of Astronomy of the University of Amsterdam, the Netherlands, in Manchester since September 2010. He then became a postdoc- 2017. After his PhD, obtained in toral researcher, and later 2013 from Scuola Normale research fellow, at the Institute Superiore (Pisa, Italy), he worked of Photonic Sciences (ICFO), as postdoc in the United States Spain. In 2018 he became Junior (University of Missouri, Group Leader of the Ultrafast Columbia) and in the Dynamics in Nanoscale Systems Netherlands (Radboud group at the Catalan Institute of University, Nijmegen). During Nanoscience and Alessandro Principi these years, he has studied the Klaas-Jan Tielrooij Nanotechnology (ICN2) in role of many-body interactions Bellaterra (Barcelona), Spain. on the transport and dynamical properties of graphene, van der His research interests are in the field of ultrafast dynamics, opto- Waals heterostructures and topological materials. He has formed electronics, nonlinear optics, light–matter interaction, and tera- successful collaborations with world-leading experimental hertz technologies, with a current focus on heat and charge trans- research teams studying, among others, hot electron relaxation in port in nanoscale materials, such as two-dimensional materials. graphene. This journal is © The Royal Society of Chemistry 2021 Nanoscale, 2021, 13, 8376–8411 | 8377 View Article Online Review Nanoscale conductivity κe. The third term describes the energy carried by the electric current density. In this term, ∏ is the local Peltier coefficient of the material, V is the local voltage, and J ¼ σð∇ V  S∇ T e Þ ð2Þ is the current density, with σ the conductivity. The Peltier coefficient is related to the local Seebeck coefficient of the material via the Onsager reciprocity relation S = ∏/Te. We will discuss the different terms of eqn (1) and (2) in detail in the This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. different sections of this review. Open Access Article. Published on 29 April 2021. Downloaded on 12/5/2022 12:11:46 PM. This review is organized as follows: In section 2. 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 eqn (1). In section 4. 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 eqn (1) and (2). We will then describe several applications enabled by hot carriers in section 6. Applications, followed by section 7. Discussion and outlook. Finally, we con- clude with section 8. Conclusion. We also provide two boxes Fig. 2 Schematic illustrations of (a) thermal excitation, (b) electrical with background information: Box 1. Boltzmann theory and Box excitation, and (c) optical excitation. 2. Experimental techniques. power25,26 or with AC power at kHz,27 MHz (ref. 28) and GHz 2 Excitation (ref. 21 and 22) frequencies. 2.1.3 Optical excitation. Optical excitation of graphene In this section we will describe different ways of exciting gra- (Fig. 2c) leads to the generation of high-energy carriers that phene, such that hot carriers are created. We then discuss, can turn into hot carriers via carrier thermalization (see from a macroscopic thermodynamic viewpoint, how a hot- Fig. 3). The absorption of light by graphene occurs through carrier distribution is established after excitation. two main processes, depending on the incident photon energy ħω with respect to the Fermi energy EF: intraband absorption 2.1 Heating sources typically dominates for ħω < 2|EF|, while interband absorption 2.1.1 Thermal excitation. The first method consists in typically dominates for ħω > 2|EF|. We will discuss light heating the graphene lattice. This creates hot carriers via elec- tron–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 gra- phene lattice locally is by using micro-heaters in the proximity of the graphene sample18,19 (see Fig. 2a). In this case, heat is created in a metallic wire that heats up due to the Joule effect upon driving a current through the micro-heater. This heating method typically generates non-uniform heating in graphene, with a temperature gradient across the graphene sheet. 2.1.2 Electrical excitation. A more direct approach for the generation of hot electrons is via Joule heating by driving a current through graphene itself (see Fig. 2b). In this case, the electron system is primarily heated, such that the electron temperature Te is typically higher than the lattice temperature TL. Electrical heating with power densities up to ∼10 kW Fig. 3 Schematic illustration of the formation of hot carriers via photo- cm−2,20–22 and even a few hundred kW cm−2 (ref. 23–25) have excitation, followed by thermalization of high-energy carriers. Since been applied to graphene, and shown to lead to an electronic high-energy carriers can also directly relax to their environment, the temperature increase ΔTe up to several thousand kelvin. branching ratio between these two processes determines the heating Electrical excitation of graphene can be achieved both with DC efficiency η. 8378 | Nanoscale, 2021, 13, 8376–8411 This journal is © The Royal Society of Chemistry 2021 View Article Online Nanoscale Review absorption in section 4. Arguably the most important feature 2.2.2 Transient heating. Transient heating refers to the of optical excitation, is that ultrashort pulses with femtose- situation where graphene is excited by a short pulse that cond duration can be used. This allows for studying the ultra- varies on a timescale faster than the heating–cooling fast dynamics of the electronic system, as well as exploiting dynamics. In the simplified case of an infinitely short input these ultrafast dynamics towards novel optoelectronic appli- pulse of power, absorption is followed by thermalization, cations. These dynamics will be the topic of section 3, where where a common temperature of the electronic system is we will discuss in more detail how light absorption leads to established. We can assign a thermalization time τtherm to carrier heating, and how subsequent cooling takes place. this process. If τtherm is shorter than the cooling time τcool, a quasi-equilibrium situation is established after absorption This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. 2.2 Macroscopic picture of carrier heating of the pulse and thermalization. In this case, we can use 2.2.1. Steady-state heating. Steady-state heating refers to Pin Open Access Article. Published on 29 April 2021. Downloaded on 12/5/2022 12:11:46 PM. the situation where graphene is excited either continuously, the peak energy density, or input fluence, Fin ¼ in order frep for example with CW light or a DC current, or by a time-depen- to calculate the peak increase in electron temperature dent power density input that varies on a timescale that is ΔTe,peak. Here, frep is the repetition rate of the incident exci- much slower than the heating–cooling dynamics of graphene tation, and the input fluence is the absorbed fluence (not charge carriers. Under these excitation conditions, a new equi- the incident fluence). A lower repetition rate gives rise to a librium situation will be established with an increased carrier larger ΔTe,peak. In order to obtain the peak temperature, we temperature Te > TL. In a simple scenario, we can ignore the ÐT solve the integral Fin ¼ TLe;peak Ce dTe , and obtain the follow- transport phenomena in eqn (1) and assume one dominant cooling channel. In this case, Q̇ext = Pin − Γcool(Te − TL), where ing sub-linear relationships between peak temperature and Pin is the absorbed power density of the external source, input fluence: Ce sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Γ cool ¼ is the heat transfer coefficient in units W m−2 K−1, 2Fin τcool Te;peak ¼ TL 2 þ for kB Te  EF ; ð6Þ and τcool is the cooling time. The heat transfer coefficient γ doped describes the flow of heat out of the electronic system, for sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi example through electron–phonon coupling. The thermoelec- 3Fin Te;peak ¼ 3 TL 3 þ for kB Te  EF : ð7Þ tric equation, eqn (1), in this simple scenario reduces to γ undoped Pin Pin τcool 2.2.3 Heating efficiency. If the thermalization process – ΔTe ¼ Te  TL ¼ ¼ : ð3Þ Γ cool Ce where high-energy carriers interact with other carriers to form In order to increase the carrier temperature Te, for a given a hot-carrier distribution – competes with other energy relax- heating power density Pin, it is beneficial to operate with a ation processes (see Fig. 3), it can be useful to define a heating small electronic heat capacity and a long cooling time. We will efficiency η. One then replaces Fin by ηFin, as done for example discuss the cooling dynamics in the next section. The elec- in ref. 33. Studies performed with relatively strong excitation tronic heat capacity, or specific heat, in two limiting cases – typically found lower heating efficiencies than studies done “doped” and “undoped” – is given by:29,30 with relatively weak excitation. For example, using an absorbed fluence in the μJ cm−2 range, a photoluminescence study 2πEF found a significantly lower Te than expected, suggesting a Ce;doped ¼ kB 2 Te ¼ γ doped Te for kB Te  EF ; ð4Þ 3ðℏvF Þ2 rather low heating efficiency.30 On the other hand, optical pump – terahertz probe studies with an absorbed fluence in 18ζð3Þ 3 2 Ce;undoped ¼ kB Te ¼ γ undoped Te 2 the nJ cm−2 range found heating efficiencies well above πðℏvF Þ2 ð5Þ 50%,34,35 demonstrating that thermalization within the elec- for kB Te  EF : tronic system dominates over alternative energy relaxation Here, ħ is the reduced Planck constant, vF the Fermi vel- channels. Below, we will discuss the microscopic processes ocity, kB Boltzmann’s constant, and ζ(3) ≃ 1.202. In the that govern the heating efficiency. “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 3 Energy dynamics operate at low Fermi energy, if the objective is to maximize the temperate increase ΔTe. Upon approaching the Dirac point, In this section, we describe the microscopic processes govern- the EF-dependence drops out. In this case, ΔTe can be further ing the energy dynamics of hot carriers. We start with a maximized by operating at low temperatures. We note that gra- thermodynamic picture of carrier heating, followed by a phene can reach record-low values of the electronic heat description of the carrier–carrier interactions that lead to ther- capacity, thus leading to large temperature rises for small exci- malization of the electronic system, after having absorbed a tation. Heat capacities as low as 2 × 10−9 J (m2 K)−1,31 and very heating pulse, for example by photoexcitation. During these recently even 1.2 × 10−11 J (m2 K)−1,32 have been found using carrier–carrier interactions the total energy is conserved, and noise thermometry measurements. the electronic system typically evolves from a non-thermal to a This journal is © The Royal Society of Chemistry 2021 Nanoscale, 2021, 13, 8376–8411 | 8379 View Article Online Review Nanoscale thermal distribution with elevated carrier temperature. We Eqn (1) can be derived by solving the equation above via then discuss the different interactions between carriers and the Chapman–Enskog method.68 Introducing the elec- other (quasi-)particles that lead to cooling of the hot-carrier tron heat capacity and other transport coefficients, the system by damping of heat to a bath. Theoretical details on resulting equation can be recast in the form of 1. calculating these scattering events can be found in Box 1. We Finally, we note that the equation above can be used to point out that the process we refer to as “thermalization” of describe also the thermalization of non-equilibrium elec- the electronic system is sometimes referred to in literature as trons (see section 3.1), not just their subsequent macro- “internal thermalization”. This is in contrast to “external ther- scopic cooling dynamics (eqn (1) – see also section 1). To malization”, which refers to thermalization of the electronic describe the relaxation of non-thermal distribution, the This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. system with the phonon system. equation above must however be solved numerically and Open Access Article. Published on 29 April 2021. Downloaded on 12/5/2022 12:11:46 PM. yields a thermal distribution on time scales controlled by strength of the electron–electron collision integral. Once Box 1 Boltzmann theory such distribution is reached, electron–electron collisions The statistical modelling of thermodynamic out-of-equi- become ineffective (their collision integral vanishes librium systems is usually performed by means of a for thermal distributions) and cooling proceeds via emis- “microscopic” Boltzmann equation, from which eqn (1) sion of phonons and can be effectively described via can be derived. The latter is commonly used to describe eqn (1). the kinetics of electrons in crystals in the so-called semi- classical regime,64,65 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) 3.1 Thermalization fields vary are much larger than the spread of wave 3.1.1 Microscopic thermodynamic picture packets themselves,66 the latter can be treated as point- Fermi–Dirac distribution. Microscopically, the electron temp- like particles with well-defined position r (the “centre” of erature Te and chemical potential μ define the shape of the the wave packet) and wavevector k. Then, as usual in stat- Fermi–Dirac distribution of a carrier system (see Fig. 1). This istical mechanics, the properties of a large collection of distribution describes the probability of finding an electron (e) particles can be determined by how they arrange in the or hole (h) with a certain energy ε, and is given by: phase space. The central object becomes the distribution function, f (r, k, t ), which determines at every instant of h i1 fe;h ðεÞ ¼ 1 þ eðε+μe;h Þ=kB Te;h : ð8Þ time t the number of particles in an infinitesimal phase- space volume element centered around the point (r, k). In equilibrium, the temperatures of the electrons in the In general, the number of particles therein changes over conduction band and the holes in the valence band are equal, time. Firstly, when external forces are applied, after an Te = Th, and so are their chemical potentials, μe = μh. The infinitesimal time dt, the particles’ position and wavevec- amount of kinetic energy density in units J m−2 in the elec- tor become r + ṙdt and k + Fdt/ħ, respectively.66,67 Here, tronic system is given by the sum of the energy density of elec- F = −e(E + ṙ × B), while ṙ is the particle velocity which, trons in the conduction band and holes in the valence band: barring Berry-curvature corrections,67 is equal to E ¼ E e þ E h . These energies are given by: ħ−1∇kεk, where εk is their energy dispersion. ð1 The number of particles in a phase-space volume E e;h ¼ εDðεÞfe;h ðεÞdε ; ð9Þ element changes over time also because of collisions, 0 described via the collision integral64 I[f (r, k, t )]. The col- where D(ε) = 2ε/(πħ νF ) is the density of states of graphene.36 2 2 lision integral accounts for all many-body scattering When a system at equilibrium, with E eq , is excited by an events a particle undergoes (against, e.g., impurities, absorbed energy density Fin, energy conservation dictates that phonons or other particles) that lead to a change of its wavevector k. After any such scattering event, a given par- E eq þ Fin ¼ E hot ; ð10Þ ticle transitions to a different phase-space volume where the “hot” state has an energy density E hot that is charac- element. Therefore, I[f (r, k, t )] is responsible for the terized by “hot” Fermi–Dirac distributions with an increased time evolution of the occupation function due to carrier temperature. The electrons and holes thus have Fermi– collisions. Dirac distributions that are broadened. This broadening has The time evolution of the distribution function, account- been observed using time-resolved ARPES measurements, after ing for both external forces and collisions, is described pulsed optical heating37–39 (see Fig. 4a). Typically, it was found by the following equation:64,65 that within 150 fs the optically excited non-thermal distri- bution thermalizes. @t f ðr; k; tÞ þ ṙ  ∇ r f ðr; k; tÞ þ ℏ1 F  ∇ k f ðr; k; tÞ A broadening of the Fermi–Dirac distribution is ¼ I½f ðr; k; tÞ: accompanied by a decrease in the chemical potential, as dic- tated by conservation of the carrier density: ne − nh = const. 8380 | Nanoscale, 2021, 13, 8376–8411 This journal is © The Royal Society of Chemistry 2021 View Article Online Nanoscale Review Time-resolved ARPES studies (see Box 2) experimentally observed that the chemical potential indeed decreases upon carrier heating37 (see Fig. 4b). The decrease of the chemical potential upon carrier heating has several implications for transport properties (see section 4 on Conductivity and section 5 on Transport) and experimentally observable quantities using the techniques described in Box 2. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. Box 2 Experimental techniques Steady-state electrical and optical measurements. A Open Access Article. Published on 29 April 2021. Downloaded on 12/5/2022 12:11:46 PM. measurement of the steady-state carrier temperature increase ΔTe can be obtained by radio-frequency Johnson noise measurements,21,22,104 taking advantage of the linear relation between the current noise spectrum and Te,105 or by detection and fitting with grey body radi- ation of the graphene hot-electron thermal emission.25,26 Another steady-state approach for the study of hot elec- trons is the excitation photomixing scheme.106 Pump-probe. In ultrafast pump–probe experiments a first light pulse, called “pump”, generates high-energy charge carriers. The temporal evolution of these photo- excited carriers is subsequently detected by measuring the absorption, transmission or reflection of a second pulse, called “probe”.107–110 The relative temporal delay Fig. 4 (a) Carrier distribution measured using time-resolved ARPES, at between the two pulses is controlled, such that the tem- three different delay times: before (blue), during (red), and after (green) poral resolution is determined by the duration of the photoexcitation, clearly indicating a broadened (hot) distribution due to ultrashort pulses. Tuning of the probe photon energy photoexcitation. The photon energy was ħω = 300 meV. (b) Extracted gives access to the energy distribution of the excited chemical potential from time-resolved ARPES measurements, as func- state. The photon energies used for the pump and probe tion of time delay after photoexcitation. Due to carrier heating, the chemical potential shifts towards the Dirac point. Panels (a) and (b) are can vary from the terahertz (ħω ∼ meV) to the UV (ħω ∼ adapted with permission from ref. 37 (Copyright 2013 Springer Nature). eV). Time-resolved ARPES. In an angle-resolved photo- emission spectroscopy (ARPES) experiment, high energy The electron and hole carrier densities, ne and nh, respectively, photons (ħω typically above 5 eV) impinge on a crystal in are given by: vacuum, ejecting electrons in free space, where their ð1 energies and exit angles are measured. These are related ne;h ¼ DðεÞfe;h ðεÞdε: ð11Þ to the energy and crystal momentum of the electrons 0 inside the sample. In the case of time-resolved ARPES, This equation relates the variation in carrier density to the the sample is first excited by an ultrashort light pulse corresponding variation in temperature and, importantly, in which generates high-energy charge carriers and sub- chemical potential. Carrier density conservation dictates the sequently the ARPES spectrum is measured with a following decrease of the chemical potential with carrier temp- second delayed high photon energy ultrashort pulse.111 erature, which in both doping regimes gives a decrease with Time-resolved photocurrent. In time-resolved photo- increasing Te: current (trPC) experiments the sample is excited by two   ultrashort laser pulses and the electrical response of the π2 kB Te 2 sample ( photocurrent or photovoltage) is measured as a μ ¼ EF 1  for kB Te  EF ; ð12Þ 6EF 2 function of their delay. If the electrical response scales nonlinearly with the incident laser intensity, such as any EF 2 μ¼ for kB Te  EF : ð13Þ hot-electron dominated photocurrent/photovoltage in 4 lnð2ÞkB Te graphene, the signal at zero/small delays will differ from We note that the Fermi energy, which is defined at absolute the signal at large delays, and the recovery dynamics of zero temperature, is related to the carrier density via36 the signal will reflect the time response of the sample.33,53,112–114 pffiffiffiffiffiffiffiffiffiffi EF ¼ +ℏvF πne;h : ð14Þ This journal is © The Royal Society of Chemistry 2021 Nanoscale, 2021, 13, 8376–8411 | 8381 View Article Online Review Nanoscale Hot-carrier density. It will be useful to consider the hot- carrier density nhot – the density of carriers with an energy larger than the chemical potential, given by† ð1 nhot;e;h ¼ DðεÞfe;h dε μe;h  2  ð15Þ 2 π 2 ¼ ðk T B e Þ þ lnð2Þμe;h B e : k T πðℏvF Þ2 12 This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. For intrinsic graphene with μe = μh = 0, this reduces to the Open Access Article. Published on 29 April 2021. Downloaded on 12/5/2022 12:11:46 PM. thermal carrier density ntherm = (π/6)(kBTe/ħνF)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 Fig. 5 (a) Intraband thermalization after photoexcitation with ħω = 1.5 this. eV, for an initial chemical potential of 0.4 eV. The density of electrons in Intraband vs. interband thermalization. We now discuss the the conduction band ne is conserved, while Te increases. (b) Interband creation of an increased hot-carrier density in the two doping thermalization after photoexcitation, for an initial chemical potential of 0.05 eV. The density of electrons in the conduction band ne, and the regimes. In the “doped” regime, with kBTe ≪ EF, only the density of holes in the valence band nh, increase, while Te = Th increases. carrier density of one band needs to be considered for energy In both cases, ne − nh is conserved, and nhot increases. These figures are and particle conservation. Thus, in the case of electron based on microscopic simulations performed in ref. 35. 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 nh, has increased. Therefore, thermalization in the case of allows for additional electron–hole pair creation across the small EF is referred to as interband thermalization. bands, at least from a thermodynamic viewpoint. We illustrate The thermodynamic picture above suggests that the valence this following ref. 35, in Fig. 5, where we show the initial and conduction bands are thermalized with each other, such Fermi–Dirac distribution directly after absorption of a heating that the holes in the valence band and electrons in the conduc- pulse of absorbed power density Pin, yet before thermalization, tion band have a common chemical potential and carrier and 300 fs later, when thermalization has occurred. These temperature. The simulations of ref. 35 show that this is an results are based on microscopic charge interactions (see Box excellent approximation when the system is examined 300 fs 1 and next subsection), yet they provide a clear thermodynamic after excitation. At shorter timescales, this picture is not always picture, with broadening of the distributions due to thermaliz- very accurate. Indeed, several experimental studies with sub-50 ation, i.e. an increased Te. They also confirm the decrease in fs time resolution have observed short-lived non-thermal chemical potential associated with thermalization. In the case distributions.39,45 A time-resolved ARPES study37 furthermore where EF is initially far away from the Dirac point (EF = 0.4 eV), showed a situation where the carrier distribution consists of this broadening only affects the electrons in the conduction two separate Fermi–Dirac distributions – one for electrons and band. The density of holes in the valence band was basically one for holes. This so-called “inverted” state was found to have zero before thermalization, and is still zero after thermaliz- a lifetime of ∼130 fs. Interestingly, the short-lived situation of ation. The density of electrons in the conduction band stays two separate distributions was only observed in the case of constant, whereas the density of hot electrons in the conduc- interband photoexcitation (with photon energy ħω > 2EF), tion band nhot,e has clearly increased. Therefore, thermaliz- whereas intraband photoexcitation (with photon energy ħω < ation in the case of large EF is referred to as intraband ther- 2EF) directly led to a single, broadened Fermi–Dirac distri- malization. In the case where EF is initially close to the Dirac bution for the valence and conduction bands. These results point (EF = 0.05 eV), the broadening affects both electrons in suggest the occurrence of a rich interplay of dynamical pro- the conduction band and holes in the valence band. Indeed, cesses during carrier thermalization. We will discuss these before thermalization there was a small density of holes in the dynamics in the following sections. valence band, whereas this density is increased after thermal- 3.1.2. Carrier–carrier scattering. Thermalization occurs pri- ization, as a result of charge and energy redistribution marily via Coulomb carrier–carrier scattering events, where between the two bands. Also the density of electrons in the carriers exchange energy and momentum by keeping their conduction band has increased. This means that – besides nhot total energy and momentum constant. We first classify carrier– – the density of interband electron–hole pairs, i.e. both ne and carrier scattering processes into scattering events within a single band, and scattering events across the bands (see † A. Tomadin, private communication. Fig. 6a). In the former case, intraband scattering occurs, which 8382 | Nanoscale, 2021, 13, 8376–8411 This journal is © The Royal Society of Chemistry 2021 View Article Online Nanoscale Review There are also scattering events that are analogous to impact ionization and Auger recombination, yet occurring within a single band. In this case, the total carrier density stays con- stant, whereas the hot-carrier density (density of carriers with energy above the chemical potential) increases or decreases, respectively. Therefore, intraband impact ionization is some- times referred to as impact excitation.16,42 We note that dyna- mical screening in the random-phase approximation48 can lead to a suppression of Auger processes in graphene,49 thus This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. favoring impact excitation and impact ionization, i.e. leading Open Access Article. Published on 29 April 2021. Downloaded on 12/5/2022 12:11:46 PM. to efficient carrier heating. Hot-carrier multiplication. In order to examine thermalization via carrier–carrier scattering more closely, we first discuss the situation where high-energy carriers have been created, typi- cally by photoexcitation of graphene with a photon energy ħω > 2EF (see Fig. 6b). This creates carriers with an energy around ε = ħω/2 above the Dirac point. In the “doped” case, where kBTe ≪ EF, these initial high-energy carriers lose kinetic energy through intraband, inelastic scattering with carriers in the Fermi sea, which gain kinetic energy. Thus, secondary hot car- Fig. 6 (a) Schematic illustration of thermalization via Coulomb-induced riers are created. This is the process of impact excitation. The carrier–carrier scattering processes after photoexcitation. Here, “CM” amount of energy Δε that is exchanged between an initial indicates interband scattering processes (impact ionization and Auger high-energy carrier and a carrier in the Fermi sea follows a dis- recombination) that can lead to the creation of multiple electron–hole tribution function that peaks around EF.42 Therefore, a single pairs per absorbed photon – carrier multiplication. Similarly, “hCM” indi- cates intraband scattering processes that can lead to the creation of ( primary) high-energy carrier can create multiple secondary multiple hot electrons per absorbed photon – hot-carrier multiplication. hot carriers during its cascade, transferring an average (b) Schematic illustration of the details of thermalization via intraband amount of Δε ≈ EF to each secondary hot carrier. This corres- carrier–carrier scattering (impact excitation), where one initial high- ponds to the creation of multiple hot carriers per absorbed energy electron leads the creation of multiple hot carriers, i.e. hot- photon, gradually increasing nhot. The occurrence of hot- carrier multiplication. In each relaxation step, the initial electron transfer an average mount of energy Δε ≃ EF to an electron in the Fermi sea. The carrier multiplication was predicted and addressed quantitat- angle φ indicates the change in momentum of the initial electron. (c) ively using optical-pump terahertz probe measurements with Calculation of the angular distribution of intraband impact excitation varying pump photon energy,34,42 and using time-resolved scattering for the initial high-energy electron, indicating near-collinear ARPES measurements.43 It was furthermore experimentally scattering. (d) Experimental observation of anisotropic carrier distri- confirmed when Wu et al. succeeded in electrically collecting bution through pump–probe spectroscopy, with pump and probe polar- izations parallel (red) or perpendicular (blue). The photon energy was ħ multiple hot-carriers per absorbed photon in a photodetector = 88 meV. Panel (a) is adapted with permission from ref. 46; panels (b) system.44 The occurrence of hot-carrier multiplication shows and (c) are adapted with permission from ref. 42 (Copyright 2013 that carrier heating in graphene is an exceptionally efficient American Physical Society); panel (d) is adapted with permission from process that can be exploited for various applications (see ref. 47 (Copyright 2016 American Physical Society). section 6). Carrier multiplication. While hot-carrier multiplication can occur in the “doped” regime, where intraband thermalization gives rise to intraband thermalization. In the latter case, inter- dominates, “real” carrier multiplication can occur close to the band scattering occurs, which gives rise to interband thermal- Dirac point, where kBTe ≫ EF. In this regime interband ther- ization. For interband processes occurring in semiconductor malization occurs, and multiple free carriers can be created materials, two important types of carrier–carrier interactions per absorbed photon. Winzer et al. showed that there is a com- are impact ionization and Auger recombination. When impact petition between the interband processes of impact ionization ionization occurs, a high-energy electron loses energy, while and Auger recombination, and predicted the occurrence of another electron with lower energy gains energy and is pro- carrier multiplication for relatively weak photoexcitation.50 moted from the valence band to the conduction band. The Carrier multiplication was observed experimentally in ref. 51, reverse process, where an electron is degraded from the con- although no unambiguous results have been presented (yet) duction to the valence band, is known as Auger recombina- showing the electrical collection of multiple electron–hole tion. These processes lead to an increase (impact ionization) pairs per absorbed photon. We note that the thermodynamic or decrease (Auger recombination) in carrier density. In gra- regimes of interband thermalization and intraband thermaliz- phene, the same interband impact ionization and Auger ation correspond to the regimes where carrier multiplication recombination processes as in semiconductors can occur, par- and hot-carrier multiplication, respectively, can occur under ticularly when the Fermi energy is close to the Dirac point. certain circumstances35,46 (see Fig. 6a). In semiconductors, the This journal is © The Royal Society of Chemistry 2021 Nanoscale, 2021, 13, 8376–8411 | 8383 View Article Online Review Nanoscale creation of multiple carriers per absorbed photon is very attrac- has occurred, at sufficiently high carrier temperatures, tive, as it enables efficient photodection and energy harvesting efficient cooling can occur, for example via optical phonon beyond the Shockley–Queisser limit.52 In graphene, hot-carrier emission (see section 3.2.1). multiplication is arguably more crucial, as there are several Momentum exchange. Carrier–carrier scattering events do not ways to exploit the additionally created hot carriers, as we will only lead to exchange of energy; also carrier momentum is discuss in section 6. modified (see angle φ in Fig. 6b). As discussed in several Effect of photon energy and power. For impact excitation, a theoretical and experimental works,42,45,47,49,50,58 carrier– higher photon energy leads to a higher energy of the primarily carrier scattering occurs preferentially in the near-collinear direc- excited high-energy carriers. Because the average energy tion, due to the kinematic constraints of carrier–carrier scatter- This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. exchange per scattering event Δε is ∼EF, a higher photon ing processes (see Fig. 6c). Purely collinear scattering, however, Open Access Article. Published on 29 April 2021. Downloaded on 12/5/2022 12:11:46 PM. energy leads to a larger number of scattering events during the is suppressed, because due to their relativistic band dispersion, relaxation cascade, and therefore to the creation of a larger electrons and holes all share the same group velocity, while back- number of secondary hot carriers per absorbed photon, as scattering is strongly suppressed because of their chirality. In low shown experimentally.34 Thermodynamically, this corresponds dimensionality, the result is a diverging duration of collisions for to a larger increase in Te per absorbed photon. Interestingly, collinear particles.59 The preferred near-collinear scattering for a given absorbed fluence Fin, the temperature increase is means that photoexcitation with linear polarization leads to an roughly independent of photon energy. This is because Fin = anisotropic distribution in momentum space, which lasts longer ħωnħω, where nħω is the absorbed photon density. This means than the interaction time with the photons (see Fig. 6d). In the that for a fixed Fin, an increase in photon energy corresponds case of excitation with a photon energy above the optical phonon to a decrease in photon density, such that the overall heating, energy, the distribution becomes isotropic in ∼150 fs.47,58 This is in principle, does not depend on photon energy for a given attributed to phonon-mediated non-collinear carrier scattering. absorbed fluence. Indeed, a constant thermal photocurrent However, in the case of excitation with a photon energy below was measured over a broad wavelength range.53 We note that a the optical phonon energy, scattering occurs purely through larger photon energy can lead to a longer duration of the ther- carrier–carrier interactions, and the anisotropy was found to malization cascade,34 which means that competing energy survive for picoseconds at 20 K. This effect can survive up to relaxation processes become more important, leading to a room temperature.47,60 lower heating efficiency η for higher photon energies. Controlling carrier–carrier scattering. We already saw that So far, we have considered the case of interband photo- changing the Fermi energy and the incident photon energy excitation, with ħω > 2EF. For intraband excitation, with ħω < changes the thermalization cascade. However, it is also poss- 2EF, the thermalization dynamics are in many ways very ible to control the intrinsic carrier–carrier scattering events. similar. Intraband excitation leads to acceleration of free car- For example, the Coulomb interaction strength decreases with riers that gain kinetic energy. These carriers then distribute temperature, leading to a longer carrier–carrier scattering their energy with the other carriers in the Fermi sea. Indeed, time at lower temperature, as observed experimentally.61 the creation of a thermalized hot state after intraband photo- Furthermore, one can control carrier–carrier scattering by excitation was observed, even within 30 fs (ref. 37) (see also proximity screening, as demonstrated recently using a high- Fig. 4a). In agreement with this observation of ultrafast ther- quality graphene sample with a nearby metal.62 Finally, the malization, there are several experimental indications, for application of a magnetic field will lead to less efficient example using mid-infrared or terahertz light,54–56 that intra- carrier–carrier scattering.63 band photoexcitation leads to efficient carrier heating, at least within the same band. 3.2. Carrier cooling When keeping the photon energy fixed, while increasing 3.2.1 Optical phonons. There are many competing cooling the incident power, or fluence, the heating efficiency η typically pathways in graphene, as summarized in Fig. 7. We first con- decreases. Indeed, both the carrier multiplication efficiency50 sider the cooling pathway, where a high-energy carrier loses and hot-carrier multiplication efficiency57 were found to energy by emitting an optical phonon. Graphene has optical decrease with increasing Pin, and time-resolved optical experi- phonons at the K-point with an energy EOP,K = 0.16 eV, and at ments that require relatively large absorbed fluences (typically the Γ-point with energy EOP,Γ = 0.2 eV.69 An important para- above 10–100 μJ cm−2), such as time-resolved photo- meter governing emission of optical phonons is the electron– luminescence30 and time-resolved ARPES37,38 typically found phonon coupling (EPC) constant, whose value has been deter- relatively low heating efficiencies. Microscopically, this is mined using first principles calculations and Raman related to the relatively large density of initial high-energy car- measurements.70–72 According to calculations in ref. 73 and riers, and relatively high carrier temperature that is being 74, carriers with an energy that is high enough to emit optical established during thermalization. This slows down the ther- phonons do so very efficiently (see Fig. 8a), with scattering malization process, thus making τheat longer. This means that times τOP well below a picosecond. Scattering of high-energy the heating efficiency η becomes smaller, due to competing carriers with optical phonons can therefore compete with energy relaxation processes for high-energy carriers, with relax- Coulomb carrier–carrier interactions that lead to thermaliz- ation time τrelax (see Fig. 3). Furthermore, once thermalization ation, and is therefore the main process determining the 8384 | Nanoscale, 2021, 13, 8376–8411 This journal is © The Royal Society of Chemistry 2021 View Article Online Nanoscale Review Time-resolved studies using relatively strong excitation, leading to many high-energy carriers, typically observed ultra- fast (few hundred fs) emission of so-called strongly-coupled optical phonons (SCOPs), cf. ref. 74–77, in agreement with the calculations of ref. 74 shown in Fig. 8a. As a result, fast ther- malization occurs not only within the electronic system, yet also between the electronic system and the optical-phonon system. This leads to charge carriers having a temperature similar to the temperature of optical phonons, while the acous- This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. tic phonon temperature is much lower. This was also observed in electronic transport studies at high electric field.25,78 Open Access Article. Published on 29 April 2021. Downloaded on 12/5/2022 12:11:46 PM. Typically, in these studies, bi-exponential cooling dynamics were observed (see Fig. 8b): sub-picosecond decay due to coup- ling to optical phonons (τOP), followed by few-picosecond Fig. 7 Schematic overview of hot-carrier cooling mechanisms. 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, yield- ing values ranging from 1.2 to 2.55 ps.30,74,79–81 For carriers with lower energy, (ε + μ) < EOP, the timescale of optical phonon emission increases exponentially (see Fig. 8a). Therefore, in the case of relatively weak excitation, or after strong excitation, followed by fast cooling to optical phonons, once Te is smaller than ∼1000 K, cooling via optical phonons was thought to be rather inefficient.33 In this case, there would not be enough high-energy carriers to emit optical phonons. Indeed, several experimental studies showed a temporary thermal decoupling between the electronic system and the phonon system, allowing the electronic system to heat up efficiently, before it cools down via phonons.82 Recently however,83,84 it was shown that optical phonons can still play an important role in hot-carrier cooling even for carrier distri- butions with a temperature close to room temperature. This is based on the idea that (at least at room temperature and above) 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 Fig. 8 (a) Calculation of scattering time between a charge carrier with carriers have relaxed by emitting optical phonons, re-thermal- a certain energy and optical phonons, τOP. (b) Experimental measure- ment of cooling dynamics using time-resolved ARPES, showing an initial ization of the electronic system occurs. This leads to “newly fast decay (∼100 fs) due to optical phonon emission. The slower tail is excited” carriers with high enough energy to relax by emitting either ascribed to optical-to-acoustic phonon conversion (hot-phonon optical phonons. These processes of optical phonon emission bottleneck) or to disorder-assisted acoustic phonon cooling. Panel (a) is and re-thermalization occur simultaneously. Thus, cooling adapted with permission from ref. 74 (Copyright 2010 American occurs by the combination of optical phonon emission and Institute of Physics); panel (b) is adapted with permission from ref. 37 (Copyright 2013 Springer Nature). continuous re-thermalization of the carrier system. A numeri- cal simulation based on this intrinsic cooling mechanism was shown to be consistent with temperature-dependent cooling heating efficiency η that governs which fraction of the dynamics obtained with optical-pump THz-probe measure- absorbed power density Pin ends up in the electronic system ments.83 More recently, it was shown that this cooling mecha- (see Fig. 3). The competition between carrier–carrier and nisms leads to bi-exponential decay, with a sub-picosecond carrier-optical-phonon scattering during the thermalization initial decay related to direct coupling to optical phonons, and phase was addressed quantitatively in ref. 34. Here, optical a few-picosecond decay due to the hot-phonon bottleneck.84 phonon emission is associated with the energy relaxation This study showed that a larger peak temperature and smaller process with timescale τrelax in Fig. 3. EF typically give rise to slower cooling dynamics for this This journal is © The Royal Society of Chemistry 2021 Nanoscale, 2021, 13, 8376–8411 | 8385 View Article Online Review Nanoscale cooling channel. For graphene where alternative cooling chan- nels (see below) are suppressed, this cooling channel ulti- mately 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.84 3.2.2 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 gov- erning acoustic phonon emission is the deformation potential, This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. whose value is obtained from first principle calculations, cf. Open Access Article. Published on 29 April 2021. Downloaded on 12/5/2022 12:11:46 PM. ref. 86. Cooling of hot carriers in graphene via momentum- conserving scattering with acoustic phonons was first described in ref. 87, where it was found that this cooling channel typically leads to lifetimes τAP on the order of nano- seconds. Indeed, it was found to be significantly less efficient than cooling via optical phonons for most peak carrier and lattice temperatures. In fact, momentum conservation and the large velocity mismatch between electrons and acoustic phonons (vF ≫ vs, where vs is the sound velocity) lead to scat- tering 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 scatter- ing, as shown in ref. 16 and 85 (see also Fig. 9a). The resultant disorder-assisted cooling mechanism, also known as “super- collision cooling”, can give rise to picosecond cooling times τSC at room temperature, especially in graphene with a sufficiently large disorder density, and associated low charge Fig. 9 (a) Schematic illustration of cooling via acoustic phonons mobility. The occurrence of this cooling mechanism was without (left) and with (right) disorder, where the presence of disorder shown experimentally using several experimental techniques speeds up the emission of acoustic phonons, also known as “supercolli- shortly after its prediction,33,88–90 as shown in Fig. 9b. These sion cooling”.85 (b) Experimental measurements of cooling dynamics using time-resolved photocurrent microscopy, together with calcu- studies showed that a lower peak temperature and a larger EF lations using disorder-assisted acoustic phonon cooling. Panels (a) and typically give rise to slower cooling dynamics for disorder- (b) are adapted with permission from ref. 33 (Copyright 2013 Springer assisted supercollision cooling to acoustic phonons. Nature). 3.2.3. Substrate phonons. Since graphene is typically placed on a substrate, and is atomically thin, carriers can relax by coupling to nearby substrate phonons, with cooling time at the moment, because encapsulation of graphene in between τsub. Low et al. theoretically addressed this cooling pathway in hBN leads to very high charge mobility: >50 000 cm2 V−1 s−1 at ref. 93. They found that cooling is an order of magnitude room temperature.96,97 It is also a naturally hyperbolic faster in the case of polar substrates, such as SiO2, than in material, with hyperbolic phonon polaritons occurring in its non-polar substrates, such as diamond. This is because of scat- two Reststrahlen bands, which are the spectral intervals tering with the surface polar phonon modes. Later, it was between the longitudinal (LO) and transverse (TO) optical shown that if a substrate hosts hyperbolic phonon modes, the phonon frequencies.98 Owing to these two spectral regions coupling between graphene hot carriers and these substrate with hyperbolic phonon polaritons, two optoelectronic experi- modes can be even stronger, leading to picosecond cooling ments have shown that cooling of hot carriers in hBN-encapsu- times at room temperature22,91,92 (see Fig. 10a). Hyperbolic lated graphene occurs via out-of-plane coupling to hyperbolic phonon polaritons exist in regions where the in-plane and out- phonon polaritons in hBN.22,91,92 These studies showed that a of-plane permittivities have opposite signs, and lead to a large higher peak temperature and a smaller EF typically give rise to photonic density of states.94 As a result, hot carriers in gra- slower cooling dynamics for super-Planckian cooling to hyper- phene can cool via near-field coupling to this large photonic bolic substrate phonon modes (see Fig. 10b). density of states. This coupling can be seen as super-Planckian 3.2.4. Discussion on cooling channels. The cooling path- radiation of the hot carriers into these optical modes of the ways via optical phonons, acoustic phonons and substrate substrate. Many layered materials are naturally hyperbolic, phonons described above, and summarized in Fig. 7, can all meaning that they have some spectral region where hyperbolic give cooling times in the ps range at room temperature. modes occur, as studied theoretically in ref. 95. Hexagonal BN However, they do have distinct dependencies on peak carrier (hBN) is arguably the preferred substrate material for graphene temperature and EF, which makes it possible to assess their 8386 | Nanoscale, 2021, 13, 8376–8411 This journal is © The Royal Society of Chemistry 2021 View Article Online Nanoscale Review Furthermore, hot-carrier photoluminescence – Planck radi- ation 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 theore- tical 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 gra- This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. phene, as observed in ref. 103. Finally, we note that under certain external conditions – beyond those that create defects Open Access Article. Published on 29 April 2021. Downloaded on 12/5/2022 12:11:46 PM. or modify the Fermi energy or peak/lattice temperature – the carrier dynamics can be altered. Most notably, the application of a magnetic was shown to lead to significantly slower dynamics.63 4 Conductivity The conductivity σ 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. 4.1 Linear conductivity There is nowadays a large amount of theoretical115–123 and experimental124,125,126,127–130 works dealing with the linear fre- quency-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 fre- Fig. 10 (a) Schematic illustration of cooling of hot carriers in graphene quencies up to a few THz the intraband conductivity domi- (red dots) via hyperbolic phonon polaritons in hBN (red lines) via near- nates, while for higher frequencies both contributions can field radiation. (b) Experimental measurements of cooling dynamics using time-resolved photocurrent microscopy, showing faster cooling play a role. Note that for graphene on a substrate the absor- for lower EF, the opposite trend compared to disorder-assisted cooling. bance (in SI units and normal incidence) is related to σ(ω) by Panel (a) is adapted with permission from ref. 91 (Copyright 2017 2Z0 1 AðωÞ ¼ 1  T ¼ Re½σðωÞ, where Z0 ¼  377 Ω is the American Physical Society); panel (b) is adapted with permission from 1 þ ns cε0 ref. 92 (Copyright 2018 Springer Nature). 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 respective contributions. What these cooling channels have in optical absorption, and it’s directly related to Joule heating as common is that they become less efficient upon lowering the it determines the resistance of a graphene sheet (see section 2). lattice temperature, as the phonon occupancy decreases. At 4.1.1 Interband conductivity. Graphene’s interband optical sufficiently low lattice temperatures – and depending on exter- conductivity can be written as:125,126,132 nal parameters, such as the spatial size of the region with hot      e2 ℏω þ 2μ ℏω  2μ carriers – cooling will be dominated by the lateral diffusion of σðωÞinter ¼ tanh þ tanh : ð16Þ hot carriers, the so-called Wiedemann–Franz cooling (see 8ℏ 4kB Te 4kB Te section 1). This cooling channel was observed using noise ther- For large enough photon energies, when ħω ≫ 2|μ| and mometry measurements, and – for their specific experiment – ħω ≫ 4kBTe, interband transitions dominate graphene’s absor- dominated for a lattice temperature below 80 K.99 It’s impor- bance and eqn (16) reduces to σ(ω)inter = σ0 = e2/(4ħ), which tant to note that this is a cooling process that is not captured thus becomes independent of frequency.124,125,133 For this by the term Γcool in eqn (1), as it is cooling that occurs as a value of the sheet conductivity the absorbance for isolated result of transport. (suspended) graphene (ns = 1) depends only on elementary We briefly mention alternative cooling channels. Coupling e2 e2 constants A ¼ π ¼ πα  2:3%. Note that α ¼ is the to graphene plasmons could play a role in the early stages of 2cε0 h 2cε0 h the energy cascade,100 and was suggested to explain the short- fine structure constant. The same “universal conductivity” can lived inverted state observed in ref. 37. Experimental signa- be obtained from the Fermi golden rule for two-dimensional tures of this cooling mechanism were observed recently.101 Dirac fermions.124 This journal is © The Royal Society of Chemistry 2021 Nanoscale, 2021, 13, 8376–8411 | 8387 View Article Online Review Nanoscale For large energies, graphene’s band structure deviates from studies, cf. ref. 34, 35, 57 and 136–139. We will now describe linear dispersion, and excitonic effects at the saddle point the three approaches, where we note that they are not mutually singularity at ∼5 eV lead to increased absorption.121,127 exclusive. Although theory predicts that the approximation of a linear Drude weight. In the first approach, the temperature depen- dispersion should hold only for relatively small values of the dence is fully captured by the so-called Drude weight:131,137 incident photon energy (ħω < 1 eV),124 experiments observed    very small deviations from the universal value of the absor- 2e2 τmr μe;h σ intra;e;h ¼ k T B e ln 2 cosh : ð18Þ bance A ∼ 2.3% for photon energies up to ħω = 1.5–2 eV.124,127 πℏ2 ð1  iωτmr Þ 2kB Te For higher photon energies, the conductivity deviates signifi- This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. cantly from the universal value. This result comes from considering spectral weight conser- Another key aspect of σ(ω)inter is that the value 2μ, which vation,137 which is related to the Thomas–Reiche–Kuhn sum Open Access Article. Published on 29 April 2021. Downloaded on 12/5/2022 12:11:46 PM. defines the crossing from intraband to interband transitions, rule for light–matter interaction.140 For the graphene conduc- can be tuned by doping and external gate voltages,134 leading tivity, this sum rule indicates that the broadening of the inter- to the possibility of designing gate-tuneable optical modu- band conductivity (see previous subsection, and Fig. 18) lators.135 According to eqn (16), due to heat-induced broaden- implies that, in the “doped” regime, the spectrally integrated ing of the Fermi–Dirac distribution and the accompanying weight of interband transitions increases. In the “undoped” shift in the chemical potential μ, interband absorption can regime, the integrated interband weight decreases. Spectral either increase (for ħω < 2EF) or decrease (for ħω > 2EF), as weight conservation then dictates that the spectrally integrated shown in Fig. 11. weight of intraband transitions should decrease (increase) for 4.1.2 Intraband conductivity. The intraband (or electrical) the “doped” (“undoped”) case. Note that the Drude weight for conductivity for electrons or holes is given by:123 graphene is different compared to conventional semi- ð   conductors or metals.127,128 This approach predicts that if e2 vF 2 1 @fe;h τmr σ intra;e;h ¼ ðω; Te Þ ¼ @εDðεÞ : ð17Þ kBTe ≪ EF, namely for weak heating and large doping, the hot- 2 0 @ε 1  iωτmr carrier conductivity is smaller than the equilibrium conduc- There are three main approaches that have been used to tivity (negative photoconductivity), which is in agreement understand how this intraband conductivity changes upon with several experiments34,35,57,136–139 (see inset Fig. 11). carrier heating, which is less straightforward than in the case Furthermore, it explains the experimental observation of an of interband transitions. Experimentally, this issue has been increase in conductivity, i.e. positive photoconductivity, close addressed in particular using optical pump – terahertz probe to the Dirac point (where the ratio Te/TF is large), as observed in ref. 35, 57, 137 and 139. This approach, however, does not take into account that momentum scattering can depend on Fermi energy and/or carrier temperature. Furthermore, the Drude weight predicts positive photoconductivity at high fluence, which is in contradiction with experiments. In ref. 137, the dependence of umr on Te was inserted phenomenolo- gically, assuming Te = TL and momentum relaxation domi- nated by phonon scattering. Sommerfeld expansion. In ref. 34, the microscopic intraband conductivity of eqn (17) was calculated in the Sommerfeld expansion regime, where the ratio 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 Fig. 11 Real part of graphene’s frequency-dependent linear optical point, as observed experimentally.34,35,57,136–139 However, it can conductivity in units of σ0 based on eqn (16) for interband transitions not be used close to the Dirac point, as the conditions for the and (18) for intraband transitions (Drude model) with fixed momentum Sommerfeld expansion are not met. relaxation time τ = 1 ps and different values of Te and μ: Te = 3 K and μ = Full microscopic approach including dynamical screening. 0.1 eV (blue curve), Te = 600 K and μ = 0.057 eV (red curve). Note that Finally, in ref. 35, the microscopic intraband conductivity eqn for interband transitions σ at Te = 600 K can either decrease or increase compared to the value at Te = 30 K close to ħω = 0.2 eV. For intraband (17) was calculated explicitly, including the energy-dependent transitions (inset) σ decreases (negative photoconductivity) as expected momentum relaxation time, governed by long-range Coulomb from the Drude model in the limit of large doping (Te < TF). scattering with impurities. Dynamical screening in the 8388 | Nanoscale, 2021, 13, 8376–8411 This journal is © The Royal Society of Chemistry 2021 View Article Online Nanoscale Review random-phase approximation was taken into account in the calculation of τmr. This approach gives positive photoconduc- tivity close to the Dirac point and negative photoconductivity away from the Dirac point, in agreement with experiments.35,57,137,139 According to ref. 35, the positive photoconductivity close to the Dirac point is the result of the increase in the charge carrier density (interband thermaliz- ation with possible carrier multiplication). Away from the Dirac point (where the charge carrier density does not This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. increase, as intraband thermalization occurs), the conductivity Open Access Article. Published on 29 April 2021. Downloaded on 12/5/2022 12:11:46 PM. is negative, because the heated carrier distribution leads to a decrease in screening of the Coulomb interaction with impurities.35 Fig. 12 Sketch of the SA process in the case of interband transitions 4.1.3 Nonlinear conductivity. The nonlinear optical pro- driven by hot-electron broadening of the Fermi–Dirac distribution and perties of graphene have been investigated extensively, both Pauli blocking. Reprinted with permission from ref. 154 (Copyright 2009 Wiley). theoretically120,122,141–144 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- than EF or the excitation photon energy, the (linear) absorp- induced changes of the refractive index, such as saturable tion is independent of the excitation intensity. However, for absorption (SA)154,155 and the Kerr effect.156 Being a centro- increasing excitation intensity and higher kBTe hot electrons symmetric crystal, graphene does not display intrinsic even- will induce Pauli blocking of interband transitions, leading to order nonlinearities in the dipole approximation. For this a nonlinear intensity-dependent saturable absorption.143,154 reason, second-harmonic generation has been observed only In the case of intraband transitions, SA is due to the photo- due to extrinsic factors, such as breaking of symmetry at an induced negative photoconductivity discussed above: an interface157 and electric currents/fields,158,159 or at large inci- increase of the excitation intensity leads to a reduction of dence angles due to the quadrupole nonlinear response.160 An σintra (inset Fig. 11) and thus to a reduction of the absorption. intuitive picture of the strong nonlinear optical response of In most materials, SA is well described by the phenomenolo- graphene can be obtained from a semi-classical model consid- gical absorption law: ering the electric field induced sheet current density within the Dirac cone.161–163 An overview of the nonlinear optical pro- aS aðIÞ ¼ þ aNS ; ð19Þ perties of graphene and related layered materials can be found 1 þ I=IS in ref. 15, while here we focus on the effect of hot carriers on the nonlinear optical sheet conductivity. where IS is the saturation intensity, while aS and aNS are the 4.1.4 Saturable absorption. Saturable absorption is a non- saturable and non-saturable absorption components.143,154 parametric third-order nonlinear optical process. The non- This form is fundamentally an exact description for a two-level linear response induced by SA corresponds to an electric field system. However, the linear band dispersion of graphene pro- oscillating at the same frequency of the incoming light, and duces a qualitatively different intensity dependence of the for this reason SA belongs to the intensity-dependent refrac- absorption, leading to the modified expression for SA:143 tive index nonlinear processes. As discussed in section 6, SA aS in graphene is widely used for passive mode-locking of ultra- aðIÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ aNS : ð20Þ 1 þ 3I=IS fast lasers thanks to its unique combination of broadband absorption and ultrafast dynamics. A detailed theoretical ana- 4.1.5 Near-IR third harmonic generation. The efficiency of lysis of SA in graphene can be found in ref. 143 and 164, nonlinear processes such as third-harmonic generation (THG) including the formal expression of the field-dependent non- is strongly enhanced due to resonant transitions. This is true linear optical conductivity σx,x(ω, E0) where x is any in-plane also for graphene, where the THG efficiency can be efficiently Cartesian direction of the graphene sheet and E0 the incom- tuned by more than one order of magnitude when in reso- ing electric field responsible for SA.164 Note that SA is poss- nance with vertical multi-photon transitions within the ible for both intraband and interband transitions143 and in Dirac cone148,149 (Fig. 13a). In addition, compared to other both cases the dynamics of the effect is dominated by the materials, graphene offers the unique possibility of tuning heating and cooling dynamics of hot electrons. A sketch of such transitions over an extremely broadband photon energy the SA mechanism in the case of interband transitions is range by applying external gate voltages and thus by tuning depicted in Fig. 12.154 The photo-excited hot electrons can EF.148 Since the THG intensity I3ω scales as I3ω ∝ |σ(3)(ω, EF, inhibit optical transitions by Pauli blocking in a range of kBTe Te)|2, THG in graphene is fully captured by the nonlinear around EF, thus reducing the absorption of photons at energy optical sheet conductivity |σ(3)(ω, EF, Te)|2, which can be micro- ħω ∼ kBTe. In the low-excitation regime when kBTe is smaller scopically calculated at Te = 0 K.141,164,165 Following symmetry This journal is © The Royal Society of Chemistry 2021 Nanoscale, 2021, 13, 8376–8411 | 8389 View Article Online Review Nanoscale 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 devi- ations from the typical cubic power dependence of THG experi- ments, as shown in ref. 150. Similar effects due to an increase of Te were observed also in FWM experiments149,153,166 and recently ultrafast all-optical modulation of THG with modu- lation depth above 90% was obtained in single-layer This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. graphene.163 Open Access Article. Published on 29 April 2021. Downloaded on 12/5/2022 12:11:46 PM. 4.1.6 THz high-harmonic generation. The case of THz high-harmonic generation (HHG) clearly illustrates the impor- tance of hot electrons in the nonlinear optical response of gra- phene. Several theoretical works predicted the existence of an intense nonlinear response of graphene at THz frequencies, based on a nonlinearity mechanism relying on coherent elec- tron motion.122,161,162,167–169 However, experiments either did Fig. 13 (a) THG in graphene is enhanced at specific values of EF corres- ponding to one-, two- and three-photon transitions within the Dirac cone. (b) The increase of Te affects the nonlinear conversion efficiency by broadening the electron distribution. The graph shows the THG efficiency, defined as I3ω/Iω, for ħω = 500 meV and a peak power density of ∼230 MW cm−2. 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:141,148 ð3Þ σ0 σ ð3Þ ðω; EF Þ ¼ i ½17Gð2jEF j; ℏωÞ 24ðℏωÞ4 ð21Þ  64Gð2jEF j; 2ℏωÞ þ 45Gð2jEF j; 3ℏωÞ; ð3Þ where σ0 = 4e4ħνF2/(32π) and G(x,y) = ln[(x + y)/(x − y)]. As dis- cussed, σ(3) has three sharp resonances at ħωm = 3EF with m = 1, 2 and 3 corresponding to one-, two- and three-photon verti- cal transitions (Fig. 13a).148,149 Interestingly, for |EF|/ħω ≪ 0.5 the THG efficiency has its minimum value although in prin- ciple all multi-photon transitions are resonant at the same time. This is because the signs of the different contributions in eqn (21) sum up to ∼0 in this limit, in analogy to the effect of quantum interference for the Raman G mode.134 Fig. 13b shows σ(3)(ω, EF, Te) 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 dramati- Fig. 14 (a) THz high-harmonic generation from single layer graphene. cally. This can be qualitatively understood considering the Te- (b) Excitation in the THz corresponds to an oscillating period of the elec- induced broadening of the Fermi–Dirac distribution. Since tric field of ∼1 ps, thus much longer than carrier–carrier scattering and comparable to the cooling time. For this reason, the nonlinear optical THG is a parametric process and the emission occurs only response of graphene in this frequency range is driven by incoherent during interaction of the electric field with the nonlinear thermodynamic effects arising from the strong temperature depen- material, all nonlinear optical experiments on graphene that dence of the sheet optical conductivity σ(ω) (inset). Panel (a) is adapted use ultrashort pulses (<100 fs) should always carefully consider with permission from ref. 56 (Copyright 2018 Springer Nature). 8390 | Nanoscale, 2021, 13, 8376–8411 This journal is © The Royal Society of Chemistry 2021 View Article Online Nanoscale Review not detect any THz HHG from multi-layer graphene,170 or only directly related in the diffusive regime, namely through the observed weak signatures of THz harmonics at 50 K.171 These Wiedemann–Franz relation: results were explained as the consequence of ultrafast e–e scat- tering (<100 fs) which suppresses the coherence in the THz- π2 2 κ¼ kB Te σ: ð23Þ induced electron velocity and current density in materials with 3e2 linear dispersion.172 It can be shown (see ref. 176) that this gives a simple Recently, HHG (Fig. 14a) with extremely high field-conver- relation between diffusivity and charge mobility μcharge, which sion efficiencies of 10−3, 10−4 and 10−5 for third, fifth and is the same result that can be obtained using the Einstein seventh harmonics, respectively, was demonstrated at room This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. relation: temperature using doped single-layer graphene.56 The result μcharge EF Open Access Article. Published on 29 April 2021. Downloaded on 12/5/2022 12:11:46 PM. was successfully interpreted based on a nonlinearity mecha- D¼ : ð24Þ nism that does not rely on coherent electron motion. Rather, 2e the THz nonlinearity was understood from a purely thermo- This predicts that the electronic heat diffusivity in graphene dynamic picture of the THz response of doped graphene. In at room temperature has a value of 100–5000 cm2 s−1, for this thermodynamic picture,56,173 hot carriers play a central typical EF up to 0.2 eV and mobilities varying from a few thou- role, because the intense nonlinear response of graphene is sand cm2 V−1 s−1 for CVD graphene up to 50 000 cm2 V−1 s−1 due to the combination of the THz-induced carrier heating for hBN-encapsulated graphene. Microscopic simulations pre- that reduces the THz conductivity,54 together with the ultrafast dicted a diffusivity of 360 cm2 s−1.177 The diffusivity of gra- heating and cooling dynamics of hot electrons (Fig. 14b). We phene was first studied experimentally using all-optical spatio- have discussed the effect of carrier heat on the THz intraband temporal scanning,178 which found a short-lived initial D of conductivity in section 4.1 and the details of the heating– 5500–11 000 cm2 s−1, followed by a value on the order of cooling dynamics in section 3. The nonlinearity arises simply 250 cm2 s−1. Recently, a spatio-temporal thermoelectric scan- because THz absorption leads to carrier heating, which leads ning microscopy experiment on hBN-encapsulated graphene to negative THz photoconductivity, i.e. reduced THz absorp- found a value of ∼2000 cm2 s−1.176 This value is in agreement tion. This effect was discussed in detail in ref. 54. Since with the measured electrical charge mobility μcharge of heating is highly efficient, this mechanism leads to very high ∼40 000 cm2 V−1 s−1 for their sample, thus being in accord- nonlinearity coefficients. The heating–cooling dynamics play a ance with the Wiedemann–Franz and Einstein relations. crucial role in observing the THz harmonics created by a mul- 5.1.2 Cooling length. An important parameter for several ticycle incident THz pulse, as detailed in ref. 56 and 173. hot-carrier-based applications is the cooling length ξcool, In the future, THz nonlinear optics with graphene could which describes how far hot charges can travel before they offer a rich playground for on-chip signal processing174 and cool down. This is given by:179 for the study of the hydrodynamic regime (see 5.1.4) with Dirac rffiffiffiffiffiffiffiffiffiffiffiffiffi fermions.175 κe τcool ξcool ¼ : ð25Þ Ce For EF ≫ kBTe, and owing to the Wiedemann–Franz relation, 5 Transport this can be written as: In this section, we address transport properties related to hot rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi τcool τmr carriers. We first discuss intrinsic properties such as hot- ξcool ¼ vF ; ð26Þ 2 carrier diffusion and thermoelectric effects. Subsequently, we discuss interlayer phenomena, involving transport of graphene where τmr is the momentum relaxation time that defines the hot carriers to a neighboring material. electrical mobility. With typical picosecond cooling times and sub-picosecond momentum relaxation times, the cooling 5.1 Diffusion length is typically below 1 μm at room temperature, as observed experimentally for example in ref. 180. Ma et al.181 5.1.1 Diffusivity. The diffusion of heat is governed by the first term on the right-hand side of the thermoelectric measured the cooling length using photocurrent microscopy, and found a non-monotonous trend with lattice temperature, equation, eqn (1). Ignoring the other terms, we obtain which they ascribed to different regimes where distinctive @Te cooling mechanisms dominated. For device configurations Ce ¼ κ e ∇ 2 Te ; ð22Þ @t where optical excitation leads to a hot-carrier distribution with where κe is the electronic part of the thermal conductivity. This a spatial size larger than 1 μm, diffusion typically plays a small equation takes the form of a standard diffusion equation with role. κe 5.1.3 Heat transfer via diffusion. Spatial diffusion of heat a thermal diffusivity D ¼ , typically given in units of cm2 Ce leads to a reduction of the average Te in the heated region. s−1. Since hot charge carriers are transporting the electronic This can be referred to as “diffusive cooling” or “Wiedemann– heat, the thermal conductivity κe and charge conductivity σ are Franz cooling”, even though it is a transport phenomenon, This journal is © The Royal Society of Chemistry 2021 Nanoscale, 2021, 13, 8376–8411 | 8391 View Article Online Review Nanoscale rather than a cooling phenomenon (associated with Γcool ). dent DC conductivity σ(ε) = en(ε)μcharge(ε). More specifically, the This cooling effect of heat transport was observed in graphene relation between the electrical conductivity and the Seebeck samples where carriers were heated by Joule heating, and coefficient is given by the semiclassical Mott formula:189 cooling occurred by diffusion into the metal leads, cf. ref. 88  and 99. Both works obtained the thermal conductivity by π2 kB 2 Te 1 @σðεÞ SMott ¼  ð27Þ measuring the temperature of their system via Johnson noise 3e σðEF Þ @ε ε¼E F thermometry measurements, and found results that were in agreement with the Wiedemann–Franz law away from the We note that this formula, which is based on Boltzmann Dirac point. These studies required cryogenic temperatures, theory (see Box 1), is derived using the Sommerfeld expansion This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. such that other cooling and momentum relaxation channels and is thus only valid in the degenerate limit kBTe ≪ EF, which is the case for most metals and doped graphene. In practice, Open Access Article. Published on 29 April 2021. Downloaded on 12/5/2022 12:11:46 PM. were suppressed. 5.1.4 Non-diffusive transport. The above derivations – with SMott can be calculated from the measurement of electrical electronic heat diffusion following Fourier’s law of diffusion – conductivity of graphene at varying EF, which is conveniently hold when the scattering mechanisms affect charge and heat achieved using a gate voltage. transport in the same way, which is usually the case. In this Since 2009, the Mott relation has been experimentally veri- situation, electronic charge and heat transport are connected fied several times in graphene.18,19,190–193 In these experi- via the Wiedemann–Franz law. In graphene, however, it was ments, the Seebeck coefficient is typically measured using shown that hydrodynamic transport can occur, where heat and microfabricated heaters to create a temperature gradient while charge flow have great similarities with the flow of a liquid in measuring the thermally induced voltage across the device a vessel, being described by the laws of hydrodynamics. (see inset Fig. 15). The first measurements of the Seebeck Several peculiar charge transport phenomena have been effect in graphene revealed a sizable thermopower value, experimentally demonstrated and explained when the system reaching ∼80 μV K−1 close to the Dirac point at room tempera- is tuned to the hydrodynamic charge transport regime, such as ture. The sign of the thermopower changes across the Dirac negative local resistance.182 This occurs when the carrier– point, as the majority carrier density switches between holes and electrons. At high carrier density, the thermopower is pro- carrier scattering length is shorter than the momentum relax- pffiffiffi ation length and shorter than the device length. portional to 1= n and increases linearly with Te, as predicted Hydrodynamic flow was also found to occur for phonons in by the Mott formula. Since the latter depends on the energy graphite, in the form of the experimental observation of wave- derivative of the conductivity, the thermopower strongly like transport of heat, called second sound.183 depends on the scattering mechanism governing the electrical A special hydrodynamic charge transport regime exists very conductivity σ. Of particular importance is how the momen- 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.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. 176. In this study, increased diffusivities of hot carriers were found in the first few hundred femtose- conds after photoexcitation with D 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. 5.2 Thermoelectric effects 5.2.1 Seebeck effect. Hot carriers in graphene can give rise to important thermoelectric and thermomagnetic phenomena.184–188 Arguably, the most prominent among them is the Seebeck effect, which refers to the conversion of a tempera- ture difference ΔTe into an electrical voltage ΔV. According to eqn Fig. 15 Measured S/Te as a function of charge carrier density n at different temperatures. The grey and purple curves represent SMott/Te (2), when J = 0, this conversion is proportional to the Seebeck and Shyd/Te, respectively. The inset shows a typical device used to coefficient, or thermopower, S = −∇V/∇Te. In a diffusive conduc- measure S, where the scale bar corresponds to 2 μm. The figure is tor, this effect originates from the net charge imbalance created adapted with permission from ref. 190 (Copyright 2016 American by the thermal diffusion of carriers that have an energy-depen- Physical Society). 8392 | Nanoscale, 2021, 13, 8376–8411 This journal is © The Royal Society of Chemistry 2021 View Article Online Nanoscale Review tum relaxation time τmr depends on carrier energy ε. As Hwang et al.194 pointed out, long-range scattering by screened charged impurities appears to be the dominant scattering mechanism in most graphene samples at room temperature. Deviations from the Mott formula were observed close to the Dirac point and at high temperature, where the Sommerfeld expansion is no longer valid. These discrepancies, as well as the nonlinear temperature dependence of S observed in clean graphene devices,195,196 can be explained theoretically by employing an effective medium This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. theory and by considering the energy-dependence of τmr for different momentum scattering mechanisms.194 Open Access Article. Published on 29 April 2021. Downloaded on 12/5/2022 12:11:46 PM. The situation is quite different in clean graphene samples with ultra-high mobility. As discussed in section 5.1.4, inelas- tic electron–electron scattering can become the dominant scat- tering process in clean graphene, and the resulting hydrodyn- amic behavior of charge carriers can lead to a violation of the Mott formula. This effect was first investigated theoretically in the context of hydrodynamic transport in Dirac fluids.197,198 It was predicted that in a purely hydrodynamic regime, the Seebeck coefficient is the entropy transported per charge of the carrier, which in the degenerate regime gives 2π2 kB 2 Te Shyd   ð28Þ 3eEF Ghahari et al. provided evidence of hydrodynamic thermo- electric transport in hBN-encapsulated graphene samples with extremely low disorder.190 At high temperature, the measured thermopower surpasses the calculated SMott, approaching the predicted Shyd (see Fig. 15). This result was explained by con- sidering the inelastic scattering between carriers, as well as the one between carriers and optical phonons. According to a recent theoretical prediction,199 this enhancement of the Seebeck coefficient in the hydrodynamic regime, combined with the low κhyd in the Fermi-liquid regime should give rise to significant improvement of the heat-to-work conversion efficiency of thermoelectric devices. 5.2.2 Photo-thermoelectric effect. 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 Fig. 16 (a) Schematic representing the generation of a PTE voltage in a response of graphene devices. In this case, light absorbed in graphene p–n junction. Light absorbed at the junction between regions graphene generates a local distribution of hot carriers charac- with different Seebeck coefficients (S1 and S2) increases the electronic terized by a temperature gradient ∇Te (see Fig. 16a). When this temperature by ΔTe and generates a photovoltage VPTE. (b) Photovoltage gradient overlaps with regions of different Seebeck coefficients as a function of back (VBG) and top (VTG) gate forming a p–n junction. (S1 and S2), a voltage VPTE is created, given by: The six-fold change of polarity of VPTE is characteristic of the PTE effect.82,179 Grey dashed lines represent lines of high resistance taken ð from transport measurements, while the white dashed line indicates VPTE ¼  SðxÞ∇ x Te dx ¼ ΔSΔTe ð29Þ where the carrier density in the two regions is equal. Panel (a) is adapted with permission from ref. 200 and panel (b) is reproduced from ref. 82 with permission from AAAS. where ΔS = 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 obtained using the Shockley–Ramo theorem201 or a general and generates a photocurrent that can be approximated as IPTE Onsager reciprocity approach,202 which account for the geo- = ΔSΔTe/R, where R is the total resistance of the device. This metrical factors (device dimensions, laser spot size, inhom- equation is valid for a rectangular device with a uniformly ogeneity) that govern the photoresponse that is measured heated junction. A more accurate estimation of IPTE can be between electrodes. This journal is © The Royal Society of Chemistry 2021 Nanoscale, 2021, 13, 8376–8411 | 8393 View Article Online Review Nanoscale The PTE effect was first observed at interfaces of single 5.3 Interlayer transport layer and bilayer graphene203 and graphene p–n junctions.82 Recent advances in the layer-by-layer assembly of 2D materials By varying EF on both side of the junction, Gabor et al.82 has led to the birth of a very active field of research on the so- showed that photovoltage exhibits a six-fold pattern (see called van der Waals heterostructures (vdWH).212–214 Fig. 16b), which is a hallmark of the PTE effect. Indeed, Song Interfacing graphene with other bulk or 2D materials has et al.179 predicted that the non-monotonic behaviour of S(EF) emerged as a powerful way to engineer material properties of shown in Fig. 15 should lead to multiple sign changes of VPTE hybrid systems. This is perhaps best exemplified by the recent (when S1 = S2 or EF ∼ 0 in both regions), in stark contrast with discovery of anomalous superconductivity in twisted bilayer the single sign reversal expected for the photovoltaic effect. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. graphene.215 These heterostructures also provide the ability to The PTE effect has also been shown to dominate the photo- control and exploit the properties of hot carriers in graphene. response of unbiased graphene-metal junctions114,204,205 and Open Access Article. Published on 29 April 2021. Downloaded on 12/5/2022 12:11:46 PM. For instance, as discussed in section 3.2.3, encapsulating gra- supported-suspended graphene interfaces.206 We note however phene using hBN gives rise to a new cooling pathway mediated that the photovoltaic and photo-bolometric effect can become by hyperbolic phonon polaritons in hBN.92 In addition to predominant in biased graphene devices.206 In section 6.1, we influencing the heat transfer, these heterostructures can discuss in more details the performance of photodetectors enable the transport of carriers in the out-of-plane direction based on the PTE and photo-bolometric effects. (see Fig. 17a). This interlayer charge transport generally takes 5.2.3 Other thermoelectric effects. In 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 eqn (1). The Peltier effect, which is the reciprocal effect of the Seebeck effect, refers to the heating or cooling that occurs at junction between regions or materials with different Peltier coefficient ∏ = STe when an electric current is flowing through it. The Peltier heat rate generated or absorbed at the junction is given by Q̇ = (∏2 − ∏1)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 gra- phene devices,196 demonstrate the advantage of using gra- phene 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.210 Finally, in the presence of an out-of-plane magnetic field Bz (with respect to the 2D graphene plane), hot carriers diffusing under ΔT are deflected by a Lorentz force, producing a trans- verse voltage Vy. This thermomagnetic effect analogous to the Hall effect is called the Nernst effect and is quantified by the Nernst coefficient ∇ Vy Sxy N¼ ¼ ð30Þ Bz ∇ x Te Bz where Sxy is the transverse component of the thermopower. The Nernst effect has been measured in graphene using micro- fabricated 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 Fig. 17 (a) Schematic of interlayer transport and intralayer thermaliz- located along the free edges of the device and which exhibits a ation of photoexcited carriers in a g/hBN/g heterostructure. (b) Band diagram of a graphene heterostructure illustrating various interlayer peak at the charge neutrality point.211 A large photo-Nernst transport mechanisms. The yellow shaded area in the hot-carrier distri- current has been observed in graphene/hBN superlattices due bution n(E) represents the carriers with an energy larger than the barrier the enhanced Nernst coefficient of the Moiré minibands, Φ that can generate thermionic emission. Panel (a) is reproduced with demonstrating the collection of multiple hot carriers.44 permission from ref. 216 (Copyright 2016 Springer Nature). 8394 | Nanoscale, 2021, 13, 8376–8411 This journal is © The Royal Society of Chemistry 2021 View Article Online Nanoscale Review place across g/X heterojunctions, where g denotes graphene junctions.240 They showed that in some cases the heat trans- and X represents a semiconducting or insulating material, and ported by thermionic emission could even dominate other involves transport mechanisms that are drastically different electronic cooling channels, resulting in PTI photodetectors from intralayer transport. Indeed, inside individual layers, with large responsivities. This photo-thermionic (PTI) effect transport is generally diffusive and carried out by the deloca- was observed soon after in g/hBN/g216 and g/WSe2/g hetero- lized carriers of the band structure. In the out-of-plane direc- structures.241 In both studies, the PTI effect is evidenced by tion, carriers are localized on each layer and their interlayer the superlinear dependence of the photocurrent on the laser transport is controlled by the potential energy barrier Φ power with photon energies below the barrier bandgap (see formed at the heterojunction. Thus, charge transport between Fig. 18a). This is a direct consequence of the exponential This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. layers typically entails processes such as tunneling and ther- thermal activation of carriers over the potential energy barrier Φ. These studies also showed that the PTI signal has a decay Open Access Article. Published on 29 April 2021. Downloaded on 12/5/2022 12:11:46 PM. mionic emission. 5.3.1 Interlayer transport mechanism. The theories of ther- time of ∼1 ps, which is consistent with the cooling time τcool mionic and field emission – also called Richardson’s217,218 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 Fig. 17b, field emission is produced by the field-induced tunneling of carriers with energy 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, gen- erating 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.223,224 Recently, several vertical transport models have been pro- posed that take into account the linear dispersion relation of graphene.223,225–228 While these models deviate from the classic Richardson’s and Fowler–Nordheim laws, the afore- mentioned 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 com- ponent 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 experi- mentally shown that this can result in a strong negative differ- ential resistance that persists up to room temperature.235,236 This coherent interlayer transport can be modelled using more Fig. 18 Competing interlayer transport processes of photoexcited car- advanced techniques such as ab initio simulations,224 and riers in a g/hBN/g heterostructure. (a) Photocurrent I as a function of nonequilibrium Green’s function237 and transfer excitation laser power P, at constant interlayer bias Vb but increasing 238 Hamiltonian formalisms. photon energies ħω (left panel), and at constant photon energy ħω but 5.3.2 Interlayer transport of photoexcited carriers. These increasing bias (right panel). The data are fitted with a power law I ∼ Pγ. interlayer transport processes also control the interlayer trans- (b) Colour map of γ as a function of Vb and ħω. The black dashed lines correspond to different tunnelling times. Regions A (γ > 1) and B (γ ∼ 1) port of out-of-equilibrium carriers in graphene hetero- indicates where the interlayer current is dominated by photo-thermionic structures. Rodriguez-Nieva et al. predicted that hot carriers emission and photon-assisted tunneling, respectively. This figure is created by photon absorption could significantly enhance ther- reproduced with permission from ref. 216 (Copyright 2016 Springer mionic emission in graphene heterostructures239 and Schottky Nature). This journal is © The Royal Society of Chemistry 2021 Nanoscale, 2021, 13, 8376–8411 | 8395 View Article Online Review Nanoscale of hot carriers in graphene. PTI emission was also observed in cally and electronic conversion efficiency of 9.8% have been g/Si Schottky junctions and was shown to lead to a significant measured,257 demonstrating the potential of graphene as a enhancement of the responsivity as compared to the metal/Si hot-carrier emitter. interface.242 By increasing the excitation photon energy or the interlayer bias voltage on their g/hBN/g devices, Ma et al.216 detected a 6 Applications second interlayer transport mechanism which they identified as photon-assisted tunneling, a process related to internal The exceptional properties of graphene, such as its broadband photoemission.243 In this transport regime, photoexcited car- absorption and ultrafast photoresponse, have led to several This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. riers in graphene tunnel through the hBN barrier before they promising applications in photonics and optoelectronics. Indeed, graphene-based devices display many desirable fea- Open Access Article. Published on 29 April 2021. Downloaded on 12/5/2022 12:11:46 PM. thermalize with other carriers. This competition between PTI emission and photon-assisted tunneling provides a way to tures, such as high-speed operation, wide spectral response, manipulate electron thermalization in graphene and to esti- gate tunability and compatibility with silicon photonics and mate the thermalization time of carriers (τheat ∼ 10 fs, see CMOS platforms. Here, we focus on photonic and opto- Fig. 18b). Both PTI emission and photon-assisted tunneling electronic devices where hot-carrier effects play a central role, were also shown to contribute to the photocurrent generation in particular photodetectors, nonlinear optical devices and in g/SiC Schottky junctions. By modelling the dependence of light emitters. the photocurrent on the laser pulse duration, a tunneling time as short as ∼0.3 fs was extracted, which offers exciting per- 6.1 Photodetectors spectives for applications in ultrafast electronics.244 Over the past decade, significant efforts have been dedicated The dynamics of interlayer charge transport in g/TMD junc- to the development of graphene-based photodetectors, as illus- tions have recently been the focus of several optical pump– trated by the many review articles published on this probe studies.245–247 While all studies have observed the gene- topic.12,13,258–262 Graphene photodetectors can convert ration of photocarriers in the TMD layer after sub-bandgap absorbed photons into an electrical signal through various photon energies, its origin is still debated. Yuan et al.246 mechanisms, including mainly the photo-thermoelectric suggested that in addition to hot carrier injection, charges in (PTE), photo-bolometric (PB), and photovoltaic (PV) effects. graphene can be photoexcited directly to the adjacent TMD While the latter is based on photoexcited carriers, PTE and PB layer via charge transfer states resulting from strong interfacial mechanisms are driven by the excess carrier temperature ΔTe. electronic coupling. In contrast, Chen et al.247 performed As discussed in section 5.2.2, PTE photodetectors produce a similar measurements and proposed that photoexcited photovoltage (or photocurrent) when hot carriers are generated charges in graphene are transferred to the TMD layer after at the junction between regions with different Seebeck coeffi- carrier intraband scattering but before electron–hole interband cients. A hot-carrier bolometer detects incident radiation by thermalization. In a recent study combining ultrafast terahertz measuring the light-induced temperature increase of the elec- and visible probes, Fu et al.245 observed a superlinear depen- tronic bath.263 This is typically achieved by measuring changes dence on the pump fluence for excitation below the TMD in the temperature-dependent resistance (dR/dT ) of the device bandgap, consistent with PTI emission observed in photo- or by measuring thermal noise. current experiments. Both PTE and PB detectors take advantage of properties of 5.3.3 Interlayer thermoelectric transport. Under constant hot carriers in graphene. As explained in section 3.1, light- thermal excitation, the interplay between heat flow and carrier induced carrier heating in graphene is efficient because the transport across the g/X interface determines the out-of-plane electron–phonon coupling is weak compared to the strong thermoelectric properties of the junction. In analogy to the in- electron–electron interactions, possibly even leading to the plane thermoelectric effect (see section 5.2.1), it is possible to generation of multiple hot carriers per incident photon. Due define an interfacial Seebeck coefficient relating the voltage to the record small heat capacity of graphene, this photo- induced by a temperature difference across the junction. In generated heat gives rise to a substantial increase in Te, this case, this thermoelectric voltage is due to thermionic which in turns generates a large PB and PTE response. emission of hot carriers over a thin barrier rather than bulk Finally, carrier cooling occurs on a picosecond timescale, diffusive transport. Using electrical heating, Seebeck coeffi- enabling high-speed photodetection. These unique thermal cients of ∼−220 and −70 μV K−1 have been measured for g/ properties, combined with the broadband photon absorption hBN interfaces248,249 and Au/g/Wse2/g/Au heterostructures,250 of graphene, make graphene-based photodetectors highly respectively. Inspired by the concept of multilayer thermionic promising for many applications. Below we concentrate on devices,251,252 several theoretical studies have proposed vdW long-wavelength and ultrafast photodetectors whose perform- heterostructures for thermionic energy conversion,228,253–256 ances are often on par with, or even better than, the best showing that they could provide better or comparable power commercially available devices, while offering specific generation and refrigeration efficiency than traditional bulk benefits. thermoelectric devices. Thermionic energy converters based 6.1.1 Long-wavelength photodetectors. The detection of on g/vacuum interfaces have also been investigated theoreti- long-wavelength photons is essential for a wide variety of appli- 8396 | Nanoscale, 2021, 13, 8376–8411 This journal is © The Royal Society of Chemistry 2021 View Article Online Nanoscale Review cations. For instance, microwave photodetectors are employed in sensitive applications ranging from radioastronomy to superconducting quantum computing, while mid- to far-infra- red (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 (i.e. large bandwidth) and high sensitivity. The This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. latter is characterized by the responsivity R, defined as the Open Access Article. Published on 29 April 2021. Downloaded on 12/5/2022 12:11:46 PM. 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 gra- phene have been demonstrated during the past decade. While these detectors directly benefit from the broadband and ultra- fast photoresponse of graphene, their intrinsic sensitivity is limited by several factors, in particular the relatively low absorption of graphene. Fig. 19 (a) Schematic of a mid-IR photodetector consisting of a hBN- One general approach to increase the sensitivity of gra- encapsulated, H-shaped graphene p–n junction with a bow-tie antenna. (b) Schematic of a graphene-based Josephson junction microwave bol- phene photodetectors is to optimize the photodetection ometer. The bolometer is embedded in a half-wave resonator to allow mechanism which converts the excess carrier temperature ΔTe DC readout (green) of the Josephson junction and microwave excitation into an electrical signal. In PTE detectors, this conversion is (blue). (c) Measured (dots) and calculated (dashed line) responsivity governed by the Seebeck coefficient S, which can be maxi- spectrum of the photodetector shown in (a) for TE-polarization. The mized by tuning the chemical potential close to Dirac point yellow shaded region corresponds to the Reststrahlen band of hBN where light absorption in graphene is enhanced due to the propagation but outside the electron–hole puddle regime (see Fig. 15. For hyperbolic phonon-polaritons. (d) NEP measured (blue dots) using the this reason, clean and electrically-tunable graphene p–n junc- device shown in (b) and the thermal fluctuation limit of the NEP tions are often used to increase the responsivity of PTE (orange). Panels (a) and (c) are reproduced from ref. 264 and panels (b) detectors.55,82,266,267 In the case of PB devices, the relatively and (d) are reproduced with permission from ref. 265 (Copyright 2020 weak dR/dT of graphene (<1%/K)268 limits the sensitivity of Springer Nature). photodetectors based on the readout of the electrical resis- tance. One way to solve this issue is to artificially increase dR/ dT, for instance, by nanostructuring graphene269 or creating a detector, resulting in an increased ΔTe. This approach has bandgap in bilayer graphene.270 Another strategy is to employ been shown to enhance the response of many hot-carrier a different scheme to read out ΔTe, for example by measuring photodetectors, from the visible to THz spectral range.279–282 the Johnson noise of graphene104,271 or the switching current Finally, another avenue to increase ΔTe is to reduce the heat of graphene-based Josephson junctions.265 transfer coefficient of hot carriers in graphene. For example, A second common strategy to increase the sensitivity of this can be accomplished by operating the photodetectors at both PTE and PB photodetectors is to increase ΔTe by enhan- low temperature181 or by suspending the graphene layer.206 cing the interaction of graphene with the incident light. This However, while this increases the responsivity of the detector, can be accomplished by integrating photonic structures, such the longer cooling time leads to a longer photoresponse time. as optical cavities, waveguides and photonic crystals, with the Several of these sensitivity-enhancement strategies have photodetector.272 For THz and mid-IR photodetectors, anten- been combined, in order to create high-performance long- nas with various geometries (see Fig. 19a) have been used to wavelength photodetectors that operate at room-temperature. compensate for the mismatch between the large area of the In the mid-IR region, where light absorption in graphene is incoming radiation and the small photoactive area of the particularly low due to Pauli blocking, Castilla et al.264 detector.55,264,273–277 At GHz frequencies, microwave resonators reported a PTE photodetectors based on clean a hBN-encapsu- (see Fig. 19b) have been used to couple light with 99% lated graphene p–n junction with a responsivity of ∼27 mA efficiency to graphene bolometers.32,265 Light–matter inter- W−1 (see Fig. 19c), an NEP of ∼80 pW Hz−0.5 and a fast rise actions can also be enhanced by taking advantage of the time 17 ns. They achieved this by designing a plasmonic polaritonic resonances of graphene or neighbouring antenna efficiently coupled to hyperbolic phonon-polaritons materials,272 such as plasmon-polariton in graphene an in hBN, which in turn concentrate mid-IR light onto the p–n nearby metal nanostructures, or hyperbolic phonon-polaritons junction. Safaei et al.280 demonstrated a plasmon-assisted PTE in hBN. The quasiparticles, which are at the center of a rich photodetector using partially nanopatterned graphene, which field of research,278 can tightly confine light in the photo- exhibits a NEP of only ∼7 pW Hz−0.5 with a response time of This journal is © The Royal Society of Chemistry 2021 Nanoscale, 2021, 13, 8376–8411 | 8397 View Article Online Review Nanoscale ∼100 ns. In the THz range, PTE photodetectors made of hBN- encapsulated graphene coupled to metallic antennas have dis- played a very low NEP (∼80 pW Hz−0.5) 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 (see Fig. 19d), down to 30 zW Hz−0.5, and a response time of 200 ns, when operating in the mK temperature range. All these This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. performance parameters are comparable or superior to other Open Access Article. Published on 29 April 2021. Downloaded on 12/5/2022 12:11:46 PM. existing long-wavelength detectors, demonstrating the poten- tial of hot-carrier photodetectors for a wide range of applications. High-speed telecom photodetectors. Graphene photodetectors also hold great potential for high-speed photodetection at telecom wavelengths, which is the key process in optical recei- vers. The latter are a critical part of optical communication systems as they often determine the overall system perform- ance.284 While most dominant technologies rely on the PV effect in semiconductors like Ge, high-speed graphene photo- detectors exploit the ultrafast dynamics of photoexcited car- riers in graphene to rapidly convert an optical signal into an electrical one. In the case of PTE and PB photodetectors, an electrical signal can be generated in less than 50 fs (ref. 53) as photoexcited carriers transfer their energy to the electronic bath, leading to the formation of a hot carrier distribution. When the light is turned off, hot carriers cool back to the lattice temperature with a characteristic time τcool ∼ 1–2 ps. This process determines the intrinsic bandwidth of the PTE and PB detectors. Early studies revealed that the intrinsic Fig. 20 (a) Schematic of a high-speed photodetector consisting of a bandwidth of graphene photodetectors can exceed 260 hBN-encapsulated graphene layer evanescently coupled to light in the buried silicon waveguide. (b) Spectral response of a PTE photodetector GHz,112,113,285 indicating their potential for data communi- measured up to 67 GHz. The dashed line indicates the 3 dB drop of the cation applications. We note, however, that the bandwidth of photoresponse. No roll off is observed in the in the frequency range of actual devices can be limited by extrinsic factors, including the the measurement. Panel (a) is reproduced with permission from ref. 286 RC time constant of the device or the bandwidth of the (Copyright 2015 American Chemical Society) and panel (b) is reproduced amplifier. with permission from ref. 287 (Copyright 2020 American Chemical Society). Several chip-integrated graphene-based photodetectors similar to the one shown in Fig. 20a have been reported during the past few years.242,286–297 In these devices, light in the waveguide is evanescently coupled to the graphene layer. up to ∼0.4 A W−1 and 12 V W−1 have been reported for PTE Compared to free-space illumination, the long interaction photodetectors. These devices typically display a bandwidth length between the guided mode and graphene increases the larger than 40 GHz (ref. 286, 289, 290 and 292) and the highest optical absorption well above the usual value of a few percent. to date surpasses 67 GHz (setup-limited, see Fig. 20b).287 Almost 100% optical absorption can be achieved with an inter- Finally, we note that graphene heterostructures, which are action length >40 μm. PB photodetectors with a bandwidth based on interlayer transport mechanisms (see section 5.3), larger than 110 GHz and a responsivity of up to ∼0.5 A W−1 are also promising for high-speed photodetection as they can have been demonstrated.295,296 However, since these photo- potentially improve the responsivity while maintaining low detectors operate with a bias voltage, they suffer from a large dark current and high bandwidth.242,294 Ultimately, improving (>100 μA) dark current, resulting in high shot noise and the large-scale fabrication298 and design13 of graphene-based increased power consumption. photodetectors could open the door to faster and more power- In contrast, PTE detectors operate at zero-bias in either efficient receivers in tele- and datacom modules. current or voltage mode, while maintaining a large photode- tection bandwidth and responsivity. Like most conventional 6.2 Nonlinear optical devices photodetectors, the current mode requires a transimpedance Owing to its strong and gate-tuneable nonlinear optical amplifier, whereas the voltage configuration can operate using response148,149 and its ease of integration in photonic devices a simpler and less costly voltage amplifier. Responsivities of such as fibers,153 waveguides152 and micro-resonators,300,301 8398 | Nanoscale, 2021, 13, 8376–8411 This journal is © The Royal Society of Chemistry 2021 View Article Online Nanoscale Review graphene has been widely explored for applications in non- rate.301,310 Several other approaches have been implemented linear optics. Notable examples are saturable absorbers and for the realization of graphene-based all-optical amplitude other optical modulators, frequency converters and modulators. With a graphene-clad microfiber (Fig. 21b) Li sensors.15,155,259,302,303 et al.299 demonstrated all-optical modulation with 38% 6.2.1 All-optical modulators. Optical modulators are funda- modulation depth and ∼2 ps switching time while Liu mental building blocks in a large variety of modern techno- et al.315 showed CW all-optical modulation with modulation logical applications: phase, frequency, amplitude and polariz- depth of 5 dB and 13 dB using single- and bi-layer graphene ation modulators are widely used in fibre optic communi- respectively. An all-optical modulator based on a Mach– cation systems, ultrafast spectroscopy, metrology, active Zehnder interferometer reached a modulation depth >50% This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. Q-switching or mode-locking of lasers and quantum infor- while maintaining an optical transmission through the device of 19%.316 All-optical modulation can be obtained also Open Access Article. Published on 29 April 2021. Downloaded on 12/5/2022 12:11:46 PM. mation. The scope of integrated optical modulators is to encode information into light with faster response time and with graphene transferred on photonic crystal cavities owing lower power consumption compared to electrical intercon- to a combination of light-induced resonance-tuning and nects. Graphene can fulfill all these requirements and, in SA.317 Finally, Ono et al.318 showed ultrafast all-optical modu- addition, can be easily integrated with standard photonic plat- lation with modulation speed of 260 fs and extinction ratio forms such as waveguides, microresonators and fibres.303 Both of 3.5 dB using only 3.5 fJ of switching energy in a graphene- amplitude135,300 and phase304 modulation at tens of GHz have loaded deep-subwavelength plasmonic waveguide. been achieved with graphene-integrated devices based on 6.2.2. Frequency converters. Similar to absorption modu- electro-optic modulation. However, the speed of electro-optic lation, in graphene also frequency conversion can be modu- modulators is limited by the capacitance of the device.13,300 lated by electric fields, as demonstrated in the case of THG148 All-optical modulation overcomes this limitation and thus fully and FWM149,152 (see section 4.1.3). Recently, ultrafast all- exploits the potential of graphene: the ultimate limit for the optical modulation with a response time of ∼2.5 ps, thus operation speed in all-optical modulators is given by the limited by the hot-electron cooling time, has also been cooling dynamics of hot electrons since the absorption and/or observed.319 Frequency conversion in the THz range is another phase modulation are defined by Pauli blocking or refractive application which, through the thermodynamic nonlinearity index changes induced by photoexcited carriers. Thus, speeds mechanism, is fully enabled by hot electrons. The attractive- well above 100 GHz are possible. ness of graphene in this field comes from its small footprint A notable example of all-optical modulation is saturable and high efficiency, as shown by observation of the record absorption (see section 4.1.3), a nonlinear and non-para- value of 1% conversion efficiency obtained with a grating-gra- metric third-order process that has been extensively used for phene metamaterial using a moderate field strength of ∼30 kV passive mode-locking of ultrafast lasers (Fig. 21a).155,303 cm−1 (ref. 174) (Fig. 21c). Graphene SAs have been used to mode-lock both solid- state305–307 and fibre lasers154,308–312 at 800 nm,307 1 μm,311 6.3 Light emitting devices 1.5 μm,154,308 2 μm (ref. 306 and 311) wavelengths and for In addition to detecting and modulating light, graphene can passive synchronization of separate laser cavities operating at also serve as a broadband and ultrafast thermal light emitter. different wavelengths.313 Saturable absorption based on σintra The superior mechanical, thermal and electronic properties of (see section 4) is possible also at THz frequencies.173,314 graphene allow it to sustain very high current densities (∼4 × Graphene SAs have enabled the generation of ultrashort 108 A cm−2),320 enabling its incandescence via Joule heating. pulses with <70 fs duration307 and lasers with GHz repetition Early studies25,78 showed that the emission spectra of electri- Fig. 21 Examples of graphene-based nonlinear devices enabled by hot electrons: (a) graphene based SA for mode-locking of fiber lasers. (b) Graphene-clad microfiber all-optical modulator. (c) Grating-graphene metamaterial for THz frequency conversion. Panels (a), (b) and (c) are adapted with permission from ref. 154 (Copyright 2009 Wiley), ref. 299 (Copyright 2014 American Chemical Society) and ref. 174 (Copyright 2020 American Chemical Society) respectively. This journal is © The Royal Society of Chemistry 2021 Nanoscale, 2021, 13, 8376–8411 | 8399 View Article Online Review Nanoscale cally biased graphene devices agree well with Planck’s law for the hot carriers into the hBN mediated by coupling to hybrid the spectral radiance U of a grey body, polaritonic modes (see section 3.2.3). Finally, another avenue to enhance Joule heating and thermal radiation in graphene is 2ðℏωÞ3 1 to confine the current flow through a narrow constriction.324 Uðℏω; Te Þ ¼ Ξ   ; ð31Þ h2 c2 ℏω Furthermore, the broad spectral emission of graphene emit- exp 1 kB Te ters can be easily tailored to meet the needs of specific where ħω is the photon energy, c is the speed of light and Ξ is applications. This can be achieved by integrating the the emissivity of the grey body. An emissivity value of ∼2.3% graphene emitter in a sub-wavelength optical structure which was measured,25,78 in agreement with Kirchhoff’s law stating modifies the electromagnetic local density of state. Different This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. that the emissivity and absorption are equal. Electronic temp- structures have been demonstrated, including photonic microcavities26,320,321,324,325 and photonic crystals.323 The Open Access Article. Published on 29 April 2021. Downloaded on 12/5/2022 12:11:46 PM. eratures Te on the order of 1500 K were measured using this technique, corresponding to a peak emission in the infrared latter enabled the strongest modulation of the black-body radi- (see Fig. 22a). Raman spectroscopy measurements further ation spectrum in the near-infrared region, which is relevant revealed that optical phonons are in thermal equilibrium with for optical communications. the electronic bath, while low-energy acoustic phonons are at a Another important advantage of graphene emitters is their significantly lower temperature. The low thermal radiation high-speed modulation rate, which is facilitated by the ultra- efficiency of these early devices (∼10−6) was attributed to the fast dynamics of hot carriers. Indeed, recent studies28,320,323 fact that Joule heat is mainly dissipated into the metallic con- reported a temporal response of ∼100 ps (see Fig. 22c) for tacts and through the SiO2 substrate via surface polar phonon various device architectures, corresponding to a modulation scattering. bandwidth of ∼10 GHz. This exceptional speed, several orders Several approaches have been investigated to improve the of magnitude faster than conventional thermal emitters, was emission efficiency of graphene emitters. To reduce vertical attributed to the rapid cooling of hot carriers, mainly due to heat dissipation, Kim et al.26 fabricated suspended graphene remote heat to the hBN or SiO2 substrates. Real-time optical emitters (see Fig. 22b) and demonstrated high thermal emis- communications at 50 Mbps was demonstrated by coupling sion in the visible range. In these devices, hot carriers with directly the emitter with an optical fiber.28 These results temperatures up to nearly 3000 K are localized at the center demonstrate the potential of graphene emitters as chip-inte- the graphene layer electronic, resulting in more efficient (∼4 × grated light sources for telecom and datacom applications. 10−3) and brighter emission. However, suspended devices must operate in vacuum since graphene quickly oxidizes at high temperature. hBN encapsulation was found to provide 7 Discussion and outlook excellent protection for graphene even at temperatures above 2000 K.320–324 These emitters have a measured efficiency of In this section we first discuss what sets apart the properties ∼1.6 × 10−5 and an estimated lifetime exceeding 4 years.320,321 of hot carriers in graphene, compared to the properties of hot These results were attributed in part to the efficient cooling of carriers in common metals. We then provide a brief compari- Fig. 22 (a) Spectral radiance of graphene under electrical bias measured as a function of the electrical power density P dissipated in the channel (dots). The measurements are fitted to Planck’s law (eqn (31), solid lines) with the indicated Te. (b) Schematic of a suspended graphene. Under electri- cal bias, light is emitted mainly from the center of the graphene sheet. (c) Generation of short (∼92 ps) light pulses from a graphene light emitter excited by a 80 ps electrical pulse, which translates into to a bandwidth of 10 GHz. Panel (a) is reproduced with permission from ref. 78 (Copyright 2010 American Physical Society). Panel (b) is reproduced with permission from ref. 26 (Copyright2015 Springer Nature). Panel (c) is reproduced with permission from ref. 320 (Copyright 2018 American Chemical Society). 8400 | Nanoscale, 2021, 13, 8376–8411 This journal is © The Royal Society of Chemistry 2021 View Article Online Nanoscale Review son with hot-carrier phenomena in other material systems cal insulators. These are material systems that are semicon- with Dirac electrons. Finally, we will mention some emerging ducting in the bulk, whereas on the surface (for 3D systems) or potential applications of hot Dirac carriers. edge (for 2D systems), protected conducting states emerge with nearly linear dispersion (see the review of ref. 337). OPTP 7.1 Graphene vs. common metals experiments on Bi2Se3 338 showed the occurrence of negative As we have shown throughout this review, hot carriers play an photoconductivity (a decrease in THz conductivity after photo- important role in the electrical and optical response of gra- excitation). This is similar to OPTP experiments on graphene, phene. We note that semiconducting two-dimensional where negative photoconductivity was interpreted as a signa- material systems can also exhibit important physical effects ture of hot carriers.34 Time-resolved ARPES measurements This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. related to hot carriers. Nevertheless, the vanishing band gap, with visible pump light performed on Bi2Se3 339,340 and Bi2Te3 341 discussed the photo-induced carrier thermalization Open Access Article. Published on 29 April 2021. Downloaded on 12/5/2022 12:11:46 PM. linear energy–momentum dispersion relation, and strong carrier–carrier interactions of the Dirac fermions in graphene and cooling dynamics. In contrast with graphene, in these set graphene apart. One important consequence of these pro- topological-insulator systems, the presence of bulk bands perties is graphene’s small electronic heat capacity Ce and heat leads to complex dynamics involving an interplay between transfer coefficient Γcool, which mainly stem from graphene’s bulk and surface charges. There have been several attempts to low charge carrier density. To emphasize this point, it is eliminate the effects of bulk charges, for example by studying instructive to examine and compare the behaviour of hot car- topological insulators excited by THz and mid-infrared light, riers in other systems. Hot carriers have been studied in such that no interband absorption takes place. For example, metals, in particular gold, for over 50 years.11,326–335 At room Luo et al. recently used this approach to demonstrate faster temperature, the electron–phonon coupling in gold leads to a dynamics for hot surface-state carriers compared to hot bulk Γcool of ∼2.2 × 1016 W m−3 K−1.336 Even when considering an carriers, for Bi2Se3 at 5 K.324 At room temperature, the experi- hypothetical layer of gold as thin as graphene, Γcool,2D ∼ 7 MW mentally observed dynamics became similar, which is due to m−2 K−1 is two orders of magnitude larger than that of gra- efficient surface–bulk coupling, as explained in ref. 339. As a phene in most conditions. For instance, for hBN-encapsulated result of the surface–bulk interactions, the precise dynamics of graphene at room temperature, the estimated Γcool mediated hot Dirac carriers in topological insulators are currently still by hyperbolic phonons is <0.1 MW m−2 K−1 for |μ| < 0.2 eV.92 under debate. Hence, under steady state conditions (see eqn (3)) and for a Another quantum material system that hosts Dirac elec- given input power, the excess electronic temperature ΔTe in trons is the 3D Dirac semimetal Cd3As2.342 Optical pump– gold is ∼100 times smaller than in graphene. For that reason, probe measurements showed clear signatures that are consist- hot carrier effects in gold are typically observed in transport ent with the formation of a hot-carrier distribution on a time- measurements only at low temperature (<100 mK), where the scale of ∼400 fs.343 Cooling was observed to be slower than in electron–phonon coupling is drastically reduced.328,332 graphene. An OPTP study showed fluence- and temperature- Alternatively, hot carriers can also be generated by short dependent cooling dynamics with two time constants – one on intense laser pulse. In this case, Te is determined by the heat the order of a few ps, and one that is up to ∼8 ps.344 These capacity coefficient γ (see eqn (6)), which in the case of gold is dynamics were interpreted within a cooling model that has 68 J m−3 K−2. Here again, the equivalent 2D value (γ2D ∼ 2 × also been used for graphene: initial fast decay by optical 10−8 J m−2 K−2) is roughly two orders of magnitude larger than phonon emission, followed by “anharmonic” phonon decay. that of graphene (<8 × 10−10 J m−2 K−2 for |μ| < 0.2 eV, see eqn Interestingly, in ref. 344 a positive THz photoconductivity was (4)), a difference which is mainly due to the large charge observed, which was also observed by Zhang et al.160 and carrier density (or Fermi energy) in gold, compared to gra- Cheng et al.163 In graphene, a positive photoconductivity was phene. This implies that for the same input fluence, the initial only found close to the Dirac point, for EF below Te;peak in graphene can be significantly larger than in gold. ∼100 meV,57,137,139 as a result of interband thermalization.35 Interestingly, these hot carriers will cool down with a similar The origin of the positive photoconductivity observed for timescale τcool = Ce/Γcool ∼ 1–2 ps.331 We also note that the Cd3As2 is still under debate. carrier heating time in metals (τheat ∼ 350–500 fs)331,334 is sig- Given these similarities in heating–cooling dynamics of nificantly longer than in graphene (<100 fs),37,45 which leads different Dirac electron systems, we could expect that similar to higher heating efficiency in the latter. All these differences technological applications as those discussed in section 6 are help explain the preponderance of hot carrier effects in gra- possible. Topological insulator materials such as Bi2Te3 have phene as compared to conventional metals. been known for their large Seebeck coefficients (see for example this review by Xu et al.345). Not surprisingly, it has 7.2 Other Dirac-electron systems been shown that significant photo-thermoelectric currents can Besides graphene, there are several other material systems be generated,346,347 similar to graphene. Furthermore, using with Dirac electrons, characterized by a linear energy–momen- the 3D semimetal Cd3As2, ultrafast, broadband photodetection tum dispersion relation. It can be expected that similar hot- based on the photo-thermoelectric effect has been demon- carrier effects as in graphene play a role in these materials. We strated.348 Also in the field of nonlinear light conversion, will discuss several quantum materials, starting with topologi- different Dirac materials have generated interesting results. This journal is © The Royal Society of Chemistry 2021 Nanoscale, 2021, 13, 8376–8411 | 8401 View Article Online Review Nanoscale Terahertz nonlinearities, for example, have been shown to orig- grant agreement GrapheneCore3 881603, the German inate from Dirac surface states in Bi2Se3,349 and, very recently, Research Foundation DFG (CRC 1375 NOA, project B5) and the THz harmonics have been generated from Cd3As2.163,350 We Daimler und Benz foundation. M. M. acknowledges support should note that in the latter case, the role of the hot carriers from the Natural Sciences and Engineering Research Council is debated, as these results were described using coherent of Canada (PDF-516936-2018) and from the Canada First carrier acceleration, rather than the thermodynamic nonlinear- Research Excellence Fund. A. P. is supported by the European ity that is responsible for the exceptionally large THz nonli- Commission under the EU Horizon 2020 MSCA-RISE-2019 pro- nearity in the case of graphene (see Fig. 14). gramme (project 873028 HYDROTRONICS). A. P. also acknowl- edges support of the Leverhulme Trust under the grant This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. RPG-2019-363. 8 Conclusion Open Access Article. Published on 29 April 2021. Downloaded on 12/5/2022 12:11:46 PM. In this review, we have described fundamental properties of hot carriers in graphene, and systems that exploit hot carriers towards useful applications with a large potential to have an References important impact on society. Whereas on a fundamental level 1 P. A. George, J. Strait, J. Dawlaty, S. Shivaraman, many physical phenomena are well understood, we conclude M. Chandrashekhar, F. Rana and M. G. 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