Non-equilibrium dynamics of photo-excited electrons in graphene: collinear scattering, Auger processes, and the impact of screening
We present a combined analytical and numerical study of the early stages (sub-) of the non-equilibrium dynamics of photo-excited electrons in graphene. We employ the semiclassical Boltzmann equation with a collision integral that includes contributions from electron-electron (e-e) and electron-optical phonon interactions. Taking advantage of circular symmetry and employing the massless Dirac Fermion (MDF) Hamiltonian, we are able to perform an essentially analytical study of the e-e contribution to the collision integral. This allows us to take particular care of subtle collinear scattering processes—processes in which incoming and outgoing momenta of the scattering particles lie on the same line—including carrier multiplication (CM) and Auger recombination (AR). These processes have a vanishing phase space for two dimensional MDF bare bands. However, we argue that electron-lifetime effects, seen in experiments based on angle-resolved photoemission spectroscopy, provide a natural pathway to regularize this pathology, yielding a finite contribution due to CM and AR to the Coulomb collision integral. Finally, we discuss in detail the role of physics beyond the Fermi golden rule by including screening in the matrix element of the Coulomb interaction at the level of the Random Phase Approximation (RPA), focusing in particular on the consequences of various approximations including static RPA screening, which maximizes the impact of CM and AR processes, and dynamical RPA screening, which completely suppresses them.
Graphene, a two-dimensional (2d) crystal of carbon atoms tightly packed in a honeycomb lattice, is at the center of an ever growing research effort, due to its potential as a platform material for a variety of applications in fields ranging from electronics, to food packaging (1); (2); (4); (5); (6); (3); (7). In particular, in optoelectronics, photonics, and plasmonics graphene has decisive advantages, such as wavelength-independent absorption, tunability via electrostatic doping, large charge-carrier concentrations, low dissipation rates, high mobility, and the ability to confine electromagnetic energy to unprecedented small volumes (8); (9); (10); (11); (12). These unique properties make it an ideal material for a variety of photonic applications (8), including fast photodetectors (13); (14), transparent electrodes in displays and photovoltaic modules (8); (15), optical modulators (16), plasmonic devices (17); (10), microcavities (18), ultrafast lasers (19), just to cite a few. Therefore, understanding the microscopic interactions between light and matter is an essential requirement to progress these emerging research areas into technological applications.
When light arrives on a graphene sample it creates a highly non-equilibrium “hot” electron distribution (HED), which first relaxes on an ultrafast timescale to a thermalized (but still hot) Fermi-Dirac (FD) distribution and then slowly cools, via optical and acoustic phonon emission, eventually reaching thermal equilibrium with the lattice. Pump-probe spectroscopy is a very effective tool to study the non-equilibrium dynamics of hot carriers and has been extensively applied to a variety of graphene samples and other carbon-based materials (20); (21); (22); (23); (24); (25); (26); (27); (28); (29); (30); (31); (32); (33); (34); (35); (11); (36); (37). There is consensus in the literature on the fact that the time scales of the thermalization process, primarily controlled by electron-electron (e-e) interactions, are extremely short, of the order of tens of femtoseconds. Indeed, early theoretical calculations (38); (39); (40); (41) based on the equilibrium many-body diagrammatic perturbation theory for an interacting system of massless Dirac Fermions (MDFs) all pointed to ultrashort e-e inelastic carrier lifetimes, with a sensitive dependence on doping.
The theory of the non-equilibrium dynamics of hot carriers in graphene has also been extensively investigated (42); (43); (44); (45); (46); (47); (48); (49); (50). Previous works, however, heavily relied on numerical analysis and did not address the following issues. When electrons in graphene are described by the low-energy 2d MDF model (2); (4); (5); (6), a special class of two-body scattering processes poses a serious conundrum. These are “collinear” events, in which incoming and outgoing momenta of the scattering particles lie on the same line (51); (52); (53); (54) (see Fig. 1). On one hand, due to the geometrical nature of these events, one is very tempted to conclude that they are irrelevant, since they lie on a one dimensional (1d) manifold embedded in a 2d space, i.e. a set of zero measure. As we will see in Sec. III.2, this intuitive statement can be formally proven by employing conservation of energy and momentum. Thus, the phase space for collinear scattering events vanishes in the case of 2d MDF bare bands. On the other hand, when e-e interactions are taken into account going beyond the single-particle picture, several interesting things happen. i) MDFs moving in a collinear way along the same directrix “spend a lot of time together” since they travel with the same speed (53), the Fermi velocity . They thus interact very strongly through the non-relativistic Coulomb interaction. A simple analysis based on the Fermi golden rule shows that this yields (52); (53); (54) logarithmically-divergent quasiparticle decay rates and transport coefficients, such as viscosities and conductivities. ii) Interactions (even at the Hartree-Fock level (55)) are responsible for deviations of the energy-momentum dispersion relation from linearity. The renormalized quasiparticle spectrum, controlled by the real part of the quasiparticle self-energy, displays a concave curvature (6), an effect that suppresses collinear scattering. iii) The broadening of the energy-momentum dispersion, which follows from the finiteness of the quasiparticle lifetime (an effect beyond the Hartree-Fock theory), opens up the phase space for collinear scattering, as thoroughly discussed in Sec. III.3. The broadening of the quasiparticle spectrum is controlled by the imaginary part of the quasiparticle self-energy, a quantity directly probed by angle-resolved photoemission spectroscopy (56); (57); (58); (59); (60); (61). iv) The situation is further complicated by the role of screening, a key phenomenon in systems with long-range Coulomb interactions (62); (63). As we will discuss in Sec. IV, static screening does not have a detrimental effect on collinear scattering. The opposite occurs when dynamical screening is considered at the level of the Random Phase Approximation (RPA). v) Non-linearities and anisotropies in the band structure beyond the MDF model (such as “trigonal warping” (2)) may affect the efficiency of screening. These issues were recently addressed in Ref. (64) by means of the equilibrium many-body perturbation theory, as we will discuss in Sec. IV.2.
All these issues raise the following question: is collinear scattering relevant or irrelevant to understand quasiparticle dynamics and transport in graphene?
Collinear (or forward) scattering plays a special role in the dynamics of quasiparticles (38) and photo-excited carriers in graphene (11). The finiteness of the quasiparticle lifetime on the mass shell (38) can be traced back to the divergence of the density of electron-hole pairs in the collinear direction. In this case, it is the only configuration in which “impact ionization” (IMI) and “Auger recombination” (AR) processes are possible (51) (see Fig. 1). IMI and AR (which we will refer to with the generic term “Auger processes”) have been studied since the later fifties (65); (66). In recent years they attracted attention in the context of semiconductors (67) and quantum dots (68); (69). IMI and AR are of fundamental interest because they strongly influence the relaxation dynamics of a HED. E.g., AR in optically-pumped 2d electron systems in the quantum Hall regime is responsible (70); (24) for emission from states with energy higher than those optically pumped, and thwarts the realization of a Landau-level laser, i.e. a laser that would operate under the 2d Landau quantization, with population inversion in the Landau levels (70). Most importantly, Auger processes can be exploited to design solar cells (71); (72) or other photovoltaic devices that can overcome fundamental limitations (73) to photocurrent production by relying on “carrier multiplication” (CM).
We reported evidence of Auger processes in graphene (11), proving the existence of IMI and CM in a short transient following ultrafast photo-excitation in the optical domain (11). The excess energy of photo-excited electrons can also be transferred to secondary electron-hole pairs by intra-band scattering, without CM from the valence to conduction band. This process, also recently experimentally demonstrated (36), proceeds by promotion of electrons from below to above the Fermi energy and does not involve processes b)-d) in Fig. 1. On the other hand, Refs. (74); (75), by probing the non-equilibrium dynamics of MDFs by time- and angle-resolved photo-emission spectroscopy, found no evidence for CM. We note, however, that Refs. (74); (75) operated in a regime of pump fluences where CM is not expected on the basis of calculations relying on static screening (48). Moreover, both experiments lacked sufficient time resolution to observe CM. Indeed, the higher the pump fluence, the shorter is the time window in which CM exists (48). E.g., for a pump fluence as in Ref. (11), CM exists in a time window (substantially larger than the time resolution in Ref. (11)). Refs. (74); (75) used much higher pump fluences, i.e. and , respectively. Ref. (74) reported strong evidence of population inversion in graphene after intense photoexcitation, similar to what reported in Ref. (76), where evidence of stimulated emission was seen for pump fluences . Because of the large fluences in Refs. (74); (76), the existence of population inversion cannot be ascribed to the absence of Auger processes.
In semiconductors, IMI (AR) creates (annihilates) an electron-hole pair and takes place when the energy transfer to (from) one electron is sufficient to overcome the band gap. Since graphene is a zero-gap semiconductor, the scattering rates of Auger processes are generally larger than in most other common semiconductors, as discussed in Ref. (51). However, Ref. (51) did not address the issue of the vanishing phase space for 2d MDF bare bands. Moreover, the IMI and AR rates calculated in Ref. (51) refer to FD distributions [Eq. (20) in Ref. (51)], thus do not apply to generic non-equilibrium situations. Finally, Ref. (51) did not discuss the role of dynamical screening, now known to play a pivotal role in the electronic and optoelectronic properties of graphene (5); (6).
Here we analyze in detail the interplay between collinear scattering and e-e interactions in the context of the non-equilibrium dynamics of photo-excited electrons.
We first show that electron lifetime effects open up a finite phase space for collinear scattering processes, thereby regularizing the pathologies mentioned above. Here we consider the broadening of the energy-momentum dispersion, but we neglect its deviations from linearity due to e-e interactions. Although these two effects could be treated in principle on an equal footing (since they are described by the imaginary and real part of the quasiparticle self-energy, respectively), changes in the dispersion due to the real part of the quasiparticle self-energy are relevant only for low carrier densities (6); (77). While our theory is general, the numerical calculations we present in Sec. V are focused on a regime with large density of photo-excited carriers, . This is a value that is typically used in experimental time-resolved techniques for mapping the relaxation dynamics of electron distributions (11); (35); (75).
We then discuss the contribution of collinear processes to the Coulomb collision integral in the semiclassical Boltzmann equation (SBE), which determines, together with electron-optical phonon (e-ph) scattering, the early stages (sub-) of the time evolution. Most importantly, we go beyond the Fermi golden rule, by introducing screening at the RPA level. Contrary to what happens in a conventional 2d parabolic-band electron gas (78); (63), the introduction of dynamical screening brings in qualitative new features. On one hand, RPA dynamical screening represents the most natural and elementary way to regularize (54) the logarithmic divergences of quasiparticle decay rates and transport coefficients (52); (53). On the other hand, due to a divergence that arises in the polarization function (41); (79); (80); (81); (82) of 2d MDFs when the collinear scattering condition is met, RPA dynamical screening completely suppresses Auger processes.
This Article is organized as follows. Sec. II describes the model MDF Hamiltonian and the SBE for the coupled dynamics of electrons and optical phonons. It also reviews the typical timescales, as set by e-e and e-ph interactions. Sec. III introduces the isotropic SBE and discusses in detail the treatment of collinear scattering in the Coulomb collision integral. The role of screening is considered in Sec. IV. Sec. V presents our main numerical results for the electron and phonon dynamics, as obtained from the solution of the isotropic SBE. Finally, Sec. VI, summarizes our main conclusions.
Ii Model Hamiltonian and the semiclassical Boltzmann equation
ii.1 MDF Hamiltonian
where the field operator annihilates an electron with 2d momentum , valley , band index (or , for conduction and valence band, respectively), and spin . The quantity represents the MDF band energy, with a slope .
MDFs interact through the non-relativistic Coulomb potential with the following 2d Fourier transform
where is an average dielectric constant (6) calculated with the dielectric constants and of the media above and below the graphene flake.
Intra-valley e-e interactions are described by the following Hamiltonian (in the eigenstate representation):
where is 2d electron system area and the delta distribution imposes momentum conservation.
The matrix element of the Coulomb potential reads
The following dimensionless coupling constant (6) controls the strength of e-e interactions (relative to the typical kinetic energy):
ii.2 Electron-electron interactions
The distribution function represents the probability that a given single-particle state with quantum numbers is occupied. The equation of motion (EOM) for this distribution function in the presence of e-e interactions is given by (83); (84); (85):
The right-hand side of the previous equation is the collision integral and the Dirac delta distributions enforce conservation of momentum and energy in each e-e scattering event. The quantity (63)
in the collision integral includes a direct (Hartree) and an exchange (Fock) term, non-vanishing if two colliding electrons have parallel spins (). This expression for the kernel in the collision integral corresponds to the second-order Hartree-Fock approximation (83). If spin-flip processes are absent (as in the case considered here), the distribution function does not depend on the spin label, which can be dropped. The summation over in Eq. (6) can be performed explicitly, obtaining the spin-independent kernel
in agreement with Ref. (44).
ii.3 Electron-phonon interactions
Electrons scatter with lattice vibrations and lose (gain) energy by emitting (absorbing) phonons. Only optical phonons in the neighborhood of the and points of the Brillouin zone (BZ) matter for electrons with energy of several hundred above the Fermi energy. At each point, both the transverse (T) and the longitudinal (L) phonon modes are considered. The distribution function of the -th phonon mode with and 2d momentum is denoted by the symbol .
The electron-phonon (e-ph) contribution to the EOM for the electron distribution is (86)
where is the area of the elementary cell of graphene’s honeycomb lattice. The terms proportional to represent electronic transitions from the single-particle state with quantum numbers , , , to the state , , . The transition is suppressed if the value of the distribution function in the final state is close to unity (Pauli blocking). The terms proportional to correspond to absorption of phonons, while the terms proportional to correspond to emission of phonons. The latter coefficient is larger than the former (Bose enhancement) because phonons, being bosonic excitations, experience bunching. The kernels can be written as:
where denotes the angle between the wave vectors and and are the electron-phonon couplings (EPCs) (87); (88); (89); (90). Phonons at the () point are responsible for intra-valley (inter-valley) scattering only.
Finally, the SBE for the phonon distribution is:
The right-hand side of the previous equation includes a phenomenological decay term which describes phonon-phonon interactions (due to the anharmonicity of the lattice). Indeed anharmonic couplings play an important role in the graphene lattice (91); (92); (93); (94); (95) and, in principle, the decay coefficient could be calculated by means of atomistic Monte Carlo simulations based on a realistic description of interatomic interactions (93); (96); (95). The decay term induces relaxation of the phonon distribution towards the equilibrium value, given by a Bose-Einstein distribution at the temperature of the lattice.
ii.4 Relaxation timescales of a hot-electron distribution in graphene
Accurate calculations of relaxation timescales in e.g. semiconductors pose a challenging problem of great theoretical and practical relevance (84); (85). In graphene, three stages of the time evolution have been identified (19); (42); (20); (21); (22); (31); (11), which follow the creation of a HED due to the action of a laser-light “pump” pulse promoting a certain density of electrons from valence to conduction band.
In the first stage, , the initial HED thermalizes to a hot FD distribution and the two bands are characterized by different chemical potentials. Recently, we were able to track this initial stage with sufficient time-resolution to directly measure the transition from a non-thermal to a hot FD distribution (11). Cooling of the hot FD distribution and equilibration of the chemical potentials between the two bands take place in the second and third stage, where the dominant process is phonon emission. The second stage, , is dominated by the emission of optical phonons (97), which in graphene are associated with an unusually large energy scale () (87); (98) and are moderately coupled to the electronic degrees of freedom. This cooling channel experiences a bottleneck when the phonon distribution heats up (97); (42). The third stage, which occurs when the bulk of the electron distribution lies below the optical-phonon energy scale, is characterized by the emission of acoustic phonons (99); (100); (19). These processes take place for , but can experience a substantial speed-up () in the case of disorder-assisted collisions (101); (102); (103). Since here we focus on the electron relaxation dynamics in the sub- time scale, we neglect the contribution of acoustic phonons in our SBE formulation.
Throughout the relaxation dynamics, phonons dissipate energy into the lattice by means of phonon-phonon interactions.
Iii Isotropic dynamics and collinear scattering processes
iii.1 Semiclassical Boltzmann equation in the isotropic limit
In this Section we simplify Eqs. (11)-(II.3) by assuming that the electron and phonon distributions are isotropic. While this assumption does not apply during the application of the pump pulse (since this couples anisotropically (44)), it has been shown that e-e interactions restore isotropy in the very short time scale of a few fs (44). Hence, our assumption applies to the thermalization and cooling stages of the time evolution, i.e. what we aim to study here.
The electron distribution is therefore assumed to depend on the wave vector only through the energy . Similarly, the phonon distribution is assumed to depend only on the magnitude of the phonon wave vector . Since the slope of the phonon dispersion is negligible with respect to , we drop the momentum dependence and use constant values and . Equations for isotropic distributions can be obtained by performing the angular integrations in the collision integrals of Eqs. (11) and (II.3).
We now outline our approach in the case of a single summation over a wave vector , involving a generic function , which depends on the direction of and another wave vector , and a functional , which depends only on the isotropic quantities and . We have (84):
where the kernel depends now only on isotropic quantities.
The calculation of the isotropic kernels in Eqs. (11) and (II.3) is summarized in the following. This approach is convenient from a computational point of view since it reduces the number of variables in the integrations that have to be carried out numerically (see App. Appendix: details of numerical calculations). Most importantly, it also allows us to handle analytically the contribution of collinear scattering to the e-e interaction in the Boltzmann collision integral.
The final results for the e-ph contributions are:
with the shorthand .
Here is the Heaviside distribution and the quantities are dimensionless. For notational convenience, we write the wave vector dependence of the e-ph kernel in the form , where and .
Finally, the e-e contribution reads
where the Coulomb kernel , with physical dimensions , represents a two-particle scattering rate. The energies of the incoming (with indexes and ) and outgoing particles (with indexes , ) are fixed. The total energy is conserved and, finally, .
iii.2 The Coulomb kernel
Here, the wave vector has modulus , while its direction can be fixed at will (i.e. along the axis) because the final result is scalar under rotations. The total wave vector is conserved in all scattering processes. In the summation over and , the Dirac delta distributions ensure that only scattering configurations that are compatible with the choice of incoming and outgoing energies are considered. Moreover, the three delta distributions restrict the 4d integral to a 1d integral (at most), after the usual continuum limit is performed. We choose to reduce the summations to an integration over the modulus of the total momentum, in terms of which we are able to represent with clarity the phase space available for Coulomb scattering (see Fig. 2).
We stress that in Eq. (20) we introduced an infinitesimal quantity in the argument of one of the delta distributions. As we will see in the next Section, if the limit is taken before calculating the 4d integral in Eq. (20), collinear scattering processes do not contribute to . We introduced to slightly relax the condition of energy conservation. The latter is recovered only in the limit . We can justify this by considering the following physical explanation. The delta distribution of conservation of energy in Eq. (6) originates from the so-called “quasiparticle approximation,” applied to the Kadanoff-Baym equations (KBEs), from which the SBE is derived (83). More precisely, the KBEs involve the true quasiparticle spectral function, which has a finite width. In the quasiparticle approximation, the spectral function is substituted with a delta distribution, which is a reasonable approximation when the width of the quasiparticle spectral function can be neglected. As we will see in this Section, in the quasiparticle approximation applied to 2d MDFs an entire class of two-body collisions (collinear processes) yields vanishing scattering rates. Our procedure takes effectively into account the fact that quasiparticles have a finite lifetime (56); (57); (58); (59); (60); (61), thereby allowing for a finite collinear scattering contribution to the Coulomb kernel. We first calculate the Coulomb kernel with a finite and then apply the quasiparticle approximation at the end of the calculation.
To make analytical progress, we now introduce elliptic coordinates for the evaluation of the Coulomb kernel (104). This is most natural because, for every fixed value of , the equation for the total energy defines a conic section in momentum space. More precisely, if (), the vector lies on an ellipse (hyperbola) with focuses located at and major axis of length . Elliptic coordinates are related to the Cartesian coordinates by the transformation , , with area element . In these coordinates, and , so that non-linear combinations between integration variables in Eq. (20) disappear. Elliptic coordinates are also extremely useful to prove that IMI and AR can only occur when are collinear (see Figs. 4a and b in Ref. (11)).
Carrying out algebraic manipulations, we rewrite Eq. (20) in the following simplified manner:
We stress that this is the most important analytical result of this Article.
Note that the integrand in Eq. (III.2) is given by the product of the kernel (II.2) and a complicated expression arising from the phase space of the e-e scattering processes, the term in the second line of Eq. (III.2). In Eq. (III.2), and the dependence of on , , , and is left implicit for the sake of simplicity. The primed sum symbol indicates summation over the available configurations of vectors , , and . To identify these configuration, one may proceed as follows. When , , , and are given, the lengths of all the sides of the two triangles and are uniquely fixed. These two triangles, which share the side of length , can be drawn on the same half-plane (with respect to a line containing the vector ) or on opposite half-planes. Thus, four geometric configurations for the wave vectors , , and are available in total. However, when all vectors are collinear, the triangles are degenerate and only one configuration is possible. Finally, in Eq. (III.2) we also introduced , , and
Let us first discuss the case in which is set to zero before carrying out the integral in Eq. (III.2). In this case, one can prove that by using the triangular and reverse triangular inequalities, . When the previous inequalities turn into equalities, the length of the vector is fixed at , , and the integration domain vanishes.
Fig. 2 plots the integration domain relative to the variable in Eq. (III.2), as a function of and , in the case . In region I, implies (if the total energy is negative, all the inequalities are reversed). All the particles are either above () or below () the Dirac point. Region I, therefore, pertains to intra-band scattering events. Similarly, one concludes that regions of type II pertain to inter-band scattering (two electrons are in opposite bands before and after the scattering). Finally, regions of type III pertain to IMI and AR. In these regions the integration domain [and, therefore, ] in Eq. (III.2) vanishes. We note that classification of regions I, II, and III holds true for arbitrary values of .
iii.3 Auger contribution to the Coulomb collision integral
We now proceed to calculate in regions of type III. In this case, a finite value of restores a non-vanishing integration domain for IMI and AR and a finite contribution to due to these processes. Note that the sign of should be chosen such that . Let us consider, for the sake of definiteness, the region of type III where . We have . The integrand in Eq. (III.2) factors into two portions, one that depends smoothly on and can therefore be evaluated at and taken out of the integral, and another singular at the boundaries of the integration domain. The integral of the latter part must be carefully evaluated and is:
Note that the result of the previous integral does not depend on , therefore remains finite in the limit .
The final result for the Auger contribution to the Coulomb kernel, valid in all regions of type III, can be written as:
where the convention for and has been introduced after Eq. (III.2). The term on the second line of Eq. (III.3) comes from the smooth portion of the integrand in Eq. (III.2). We stress that Eq. (III.3) follows from the general expression (III.2) without a priori restrictions to collinear scattering configurations. Although Eq. (III.3) mathematically coincides with Eq. (14) of Ref. (51), Ref. (51) does not report any discussion on how to bypass the vanishing phase space problem for 2d MDFs. Here, the finiteness of IMI and AR contributions to the Coulomb kernel, as for Eq. (III.3), originates from electron-lifetime effects. Incidentally, since the value of the integral in Eq. (23) does not depend on the value of , the precise mechanism (e-e interactions, electron-impurity scattering, etc.) responsible for the broadening of the delta distribution in Eq. (6) into a finite-width quasiparticle spectral function is unimportant. Finally, we emphasize that IMI and AR scattering rates were calculated in Ref. (51) for FD distributions only, as seen in Eq. (20) of Ref. (51). On the contrary, Eqs. (III.2) and (III.3) can be used to solve the SBE (11) with arbitrary non-equilibrium initial conditions.
iii.4 Logarithmically divergent collinear scattering rates
We finally consider in regions of type I and II. The integrand in Eq. (III.2) diverges as for , which coincides with the upper or lower boundaries of the integration domain for regions of type I and II. When , intra-band or inter-band scattering occur in a collinear fashion. This strong divergence of the integrand physically arises from the expression for the phase space of e-e scattering, while is well behaved. Therefore, diverges for both intra-band and inter-band scattering processes. A possible way to cure this divergence (104); (52); (53) is to introduce an ultraviolet cut-off , which yields a Coulomb kernel . Logarithmic enhancements for 2d Fermions with a linear dispersion were discussed in Ref. (104), and allow one to find a SBE solution in the form of an effective equilibrium distribution, with parameters depending on the direction of motion (53). Peculiar properties of MDFs, which are sensitive to collinear scattering, include a finite conductivity in the absence of impurities (52) and an unusually low shear viscosity (105).
A different way to treat this divergence is to invoke screening, which suppresses the kernel and regularizes the behavior of the integrand in a neighborhood of . This approach is discussed in the next Section in great detail.
Iv Going beyond the Fermi golden rule: the role of screening
The SBE is a second-order expression in the bare Coulomb potential and describes e-e interactions at level of the Fermi golden rule (63). This approximation neglects many-body effects, and most importantly electronic screening. Formally, screening can be taken into account (63) by substituting the bare Coulomb potential with a screened potential . When the 2d MDF system is out of equilibrium, the screening properties change in time and the screened potential depends on time as well.
It has been pointed out (11); (106); (107) that screening may preempt the strong collinear scattering singularity mentioned above and suppress Auger processes. Indeed, the RPA dynamical dielectric function at equilibrium diverges for collinear scattering configurations for which (see Sec. IV.1). When substituted into Eq. (III.2), the screened potential vanishes like , thereby compensating the aforementioned divergence arising from the expression of the phase space. The integral in Eq. (III.2) is then finite, while the contribution to the Coulomb kernel due to Auger processes vanishes.