We theoretically examine the effect of carrier-carrier scattering processes on the intraband radiation absorption and their contribution to the net dynamic conductivity in optically or electrically pumped graphene. We demonstrate that the radiation absorption assisted by the carrier-carrier scattering is comparable with Drude absorption due to impurity scattering and is even stronger in sufficiently clean samples. Since the intraband absorption of radiation effectively competes with its interband amplification, this can substantially affect the conditions of the negative dynamic conductivity in the pumped graphene and, hence, the interband terahertz and infrared lasing. We find the threshold values of the frequency and quasi-Fermi energy of nonequilibrium carriers corresponding to the onset of negative dynamic conductivity. The obtained results show that the effect of carrier-carrier scattering shifts the threshold frequency of the radiation amplification in pumped graphene to higher values. In particular, the negative dynamic conductivity is attainable at the frequencies above 6 THz in graphene on SiO2 substrates at room temperature. The threshold frequency can be decreased to markedly lower values in graphene structures with high-κ substrates due to screening of the carrier-carrier scattering, particularly at lower temperatures.
© 2014 Optical Society of America
Graphene, a two-dimensional carbon crystal, possesses no energy band gap and, hence, is promising for detection and generation of far-infrared and terahertz (THz) radiation [1–3]. Several concepts of graphene-based THz lasers with optical pumping  or electrical (injection) pumping [5,6] have been proposed and analyzed. Recently, the possibility of THz-wave amplification by optically pumped graphene was shown experimentally .
Generally, one of the main reasons for low efficiency of THz semiconductor laser is the intraband (Drude) radiation absorption , which grows rapidly with decreasing the radiation frequency. This absorption process aggressively competes with the radiation amplification due to the stimulated interband electron transitions under the conditions of population inversion. This problem is crucial for graphene-based lasers as well.
The optical conductivity of clean undoped graphene is equal to the universal value σq = e2/4h̄, where e is the elementary charge and h̄ is the Planck’s constant. This value corresponds to the interband radiation absorption coefficient πα = 2.3 % , where α = e2/h̄c is the fine-structure constant (c is the speed of light in vacuum). The optical conductivity of pumped graphene can be negative due to prevailing stimulated electron transitions from the conduction to the valence band associated with the interband population inversion . The interband dynamic conductivity appears to be negative at frequencies ω < 2εF/h̄, where εF is the quasi-Fermi energy of pumped carriers [see Fig. 1(a) and 1(b)]. However, the coefficient of radiation amplification in pumped graphene is still limited by 2.3 %. Hence, for the operation of graphene-based lasers the Drude absorption coefficient  should lie below 2.3 %. In the previous models of negative dynamic conductivity in pumped graphene, it was usually assumed that the Drude absorption originates from electron and hole scattering on impurities, lattice defects, and phonons [Fig. 1(d)], which can, in principle, be removed almost completely in high-quality graphene samples and at low temperatures (see, for example, [3,13]). However, there is an unavoidable mechanism leading to the intraband absorption, namely, the radiation absorption assisted by the carrier-carrier scattering [Fig. 1(c)].
The carrier-carrier scattering (for brevity denoted as c-c scattering) was shown to be a key factor in the relaxation kinetics of photoexcited electrons and holes in graphene [7, 9, 14, 15]. It can also be responsible for weakly temperature-dependent minimal dc conductivity of graphene [16–20]. The strength of c-c scattering in graphene is governed by the relatively large coupling constant, αc = e2/(κ0h̄vF) ∼ 1, where vF = 106 m/s is the velocity of massless electrons and holes (Fermi velocity) and κ0 is the background dielectric constant.
Among a variety of phenomena originating from the carrier-carrier scattering in graphene-based structures, there are modifications of quasiparticle spectra , relaxation [14, 15], thermalization, and recombination . In this paper, we focus on the scattering-assisted intraband radiation absorption. The latter appears to be a quite strong absorption mechanism, especially in the pumped graphene with population inversion. We derive the expression for the real part of the intraband dynamic conductivity Reσcc (which is proportional to the intraband contribution to the radiation absorption) arising due to carrier-carrier collisions in clean pumped graphene. The probability of corresponding intraband radiation absorption process is evaluated using the second-order perturbation theory and the Fermi golden rule. Comparing Reσcc with the inter-band conductivity Reσinter, we find the threshold values of the quasi-Fermi energies of pumped carriers, required to attain the net negative dynamic conductivity (negative absorption coefficient) at given frequency, and calculate the pertinent threshold frequencies. We also show that screening of the Coulomb potential by the carriers in graphene plays the significant role in the c-c scattering. Because of screening, the intraband conductivity due to the c-c collisions grows slowly (almost linearly) with increasing the quasi-Fermi energy of pumped carriers. We find that in the pumped graphene, the main contribution to the intraband conductivity arises from electron-hole (e–h) scattering, while the electron-electron (e-e) and hole-hole (h-h) collisions yield less than one tenth of its total value. The rate of c-c scattering and the corresponding radiation absorption can be reduced in graphene clad between materials with high dielectric constants.
In the present paper, we shall restrict our consideration of dynamic conductivity within terahertz range (1–15 THz). At higher frequencies, the optical response is governed mainly by interband transitions  which lead to the absorption of radiation at h̄ω > 2εF, while the Drude absorption drops proportionally to ω−2. At lower frequencies, one can readily use the static limit of Boltzmann / hydrodynamic theory to obtain the conductivity.
The paper is organized as follows. In Sec. 2, we derive the general equations for the radiation power absorbed due to c-c scattering-assisted intraband transitions, and find the pertinent contribution to the dynamic conductivity. In Sec. 3, we obtain the dependences of the intraband, interband, and net dynamic conductivity on the radiation frequency, the quasi-Fermi energy, the background dielectric constant; and find the frequency threshold of the radiation amplification. Section 4 deals with discussion of the obtained results and their validity. In Sec. 5, we draw the main conclusions. Some mathematical details are singled out in Appendix.
2. Intraband dynamic conductivity general equations
There are two mechanisms through which the c-c scattering affects the optical conductivity, and, hence, absorption of radiation. First, the finite lifetime of carriers leads simply to the smearing of the interband absorption/amplification edge . Second, a free electron can absorb a photon and transfer the excess energy and momentum to the other carrier [Fig. 1(c)]. The latter mechanism is quite similar to the conventional Drude absorption, where the excess momentum is transferred to an impurity or phonon [Fig. 1(d)]. However, in the case of graphene, the radiation absorption due to c-c scattering exhibits a number of unusual features.
In semiconductors with parabolic bands, e-e scattering cannot directly affect the conductivity and absorption of radiation. The reason is that the total current carried by two electrons is not changed in the scattering process due to momentum conservation. In graphene, however, electron velocity and momentum are not directly proportional to each other (sometimes referred to as “momentum-velocity decoupling” ). Hence, the momentum conservation does not imply the current conservation, and e-e scattering can contribute to relaxation of current  and optical absorption. Such effects were studied in III-V semiconductors in the light of p4-corrections to the parabolic bands . In graphene, the carrier dispersion law is not parabolic ”from the very beginning”. Hence, those effects can play a more considerable role.
Electron-hole (e–h) scattering processes do not conserve the total current and can contribute to radiation absorption independent of the carrier spectrum. Apart from graphene with Fermi energy close to Dirac point, e–h scattering plays a minor role in conductivity due to vanishing number of holes. This is not the case of the pumped graphene, where e–h scattering is intensified due to the great number of carriers in both bands.
We consider a two-step quantum-mechanical process: the quasiparticle (electron or hole) absorbs light quantum passing to the virtual state, then it collides with other quasiparticle and transfers the excess energy (and momentum) to it. If the frequency of electromagnetic wave ω substantially exceeds the electron (or hole) collision frequency ν, one can use the Fermi golden rule to calculate the absorption probability in unit time. We will first consider the radiation absorption due to e-e collisions, and then generalize the result to account for e–h processes. The initial momenta of incoming quasiparticles are denoted by p1 and p2, while the final momenta are p3 and p4.
To work out the matrix elements appearing in the Fermi golden rule we apply the second order perturbation theory considering the two perturbations. The first one, V̂F (t), is the interaction of electron with external electromagnetic field
The second perturbation, V̂C, is the Coulomb interaction between carriers. It has nonzero matrix elements connecting the two-particle states |p1, p2〉 and |p3, p4〉= |p1 + q, p2 − q〉, where q is the transferred momentum. We denote these matrix elements by VC(q) ≡ 〈p3, p4|V̂C|p1, p2〉,25].
A well-known relation for the second-order matrix element connecting initial |i〉 = |p1, p2〉 and final |f〉 = |p3, p4〉 states reads 
Equation (6) clearly demonstrates that for parabolic bands no absorption due to e-e scattering can occur as vp1 + vp2 − vp3 − vp4 turns to zero as a result of momentum conservation.
The power, P, absorbed by an electron system is calculated using the Fermi golden rule, taking into account the occupation numbers of the initial and final states:Eq. (7) is the electron distribution function, which is assumed to be the quasi-equilibrium Fermi function: Eq. (7) we have also neglected the exchange-type scattering, which occurs only between electrons with same spin and from the same valley. This assumption does not significantly affect the final numerical values.
Finally, to obtain the real part of the intraband dynamic conductivity due to e-e collisions Reσee, we express the vector-potential in Eq. (6) through electric field and equate (7) with . As a result, this conductivity is expressed via the universal optical conductivity of clean graphene σq, the coupling constant αc = e2/(κ0h̄vF), and the dimensionless ’collision integral’ Iee,ω:
At the frequencies exceeding the carrier collision frequency ω ≫ ν, the contributions to the real part of dynamic conductivity from different scattering mechanisms are summed up. The intraband conductivity due to e-h collisions is given by expression similar to (9–10), with several differences. First, the change in current carried by electron and hole is given by Δneh =(k1+/k1+ − k2−/k2−) − (k1−/k1− − k2+/k2+) as the charge of hole is opposite to that of electron. Second, along with ’simple’ electron-hole scattering, the annihilation-type interaction between electron and hole is also possible. In such process, electron and hole annihilate, emit a virtual photon, which produces again an electron-hole pair. The probability of such process is the same as of scattering process, but the overlap factors should be changed to . As a result, the electron-hole scattering contribution to the optical conductivity becomes
Accordingly, the net intraband conductivity due to c-c collisions Reσintra in symmetrically pumped graphene isEq. (13) timing it by 4π/c. The dimensionless ’collision integrals’ Iee,ω and Ieh,ω are given by Eqs. (10) and (12). They can be evaluated numerically passing to the elliptic coordinates . In those coordinates, the delta-function can be analytically integrated, reducing the dimensionality of integral by unity. This procedure is described in Appendix.
3. Analysis of intraband and net dynamic conductivity
It is natural that the obtained expression (13) for the real part of conductivity is proportional to the universal ac conductivity of clean undoped graphene σq, and the square of the coupling constant αc. The latter characterizes the strength of Coulomb interaction between carriers in graphene. The frequency behavior of Reσcc follows the well-known Drude-like dependence ω−2. At frequencies h̄ω ≪ kBT, the pre-factor before the collision integral in Eq. (13) yields the ω−2-dependence. At large frequencies h̄ω ≥ kBT, the pre-factor behaves as ω−3, but the ”collision integrals” Iee,ω and Ieh,ω grow linearly due to the increasing number of final states.
In what follows, we shall study the dynamic conductivity of sufficiently clean pumped graphene. In this situation, the positive intraband conductivity is due to carrier-carrier collisions only, thus Reσintra ≡ Reσcc. Figure 2 shows the dependences of Reσintra given by Eq. (13) (normalized by σq) on frequency ω/2π for the symmetrically pumped graphene at different quasi-Fermi energies εF and different background dielectric constants κ0 at room temperature T = 300 K. In Fig. 2, we also show the frequency-dependent real part of the dynamic conductivity associated with the interband transitions :
From Fig. 2 one can see that at frequencies above ∼ 6.5 THz at κ0 = 1 (and above ∼ 2.5 THz at κ0 = 20) the Drude conductivity due to c-c collisions lies below σq. Accordingly, the net conductivity Re(σintra + σinter) can be negative in this frequency range at some level of pumping.
As the background dielectric constant κ0 is increased, the intraband conductivity due to cc collisions drops, which is illustrated in Fig. 2. Ex facte, one could expect that it scales as , and increasing the dielectric constant one could reduce the radiation absorption due to c-c collisions almost to zero. Such considerations are actually irrelevant due to screening. The Thomas-Fermi screening momentum in pumped graphene is given by
In Fig. 3 we show the net dynamic conductivity including interband and intraband c-c contributions. At weak pumping (εF ≃ 20 meV) the negative dynamic conductivity is attainable only in graphene clad between high-κ substrates. At elevated pumping (εF ≃ 40 meV) the negative conductivity is possible at any value of κ0. Further increase in the pumping level is not much efficient as the interband conductivity reaches its maximum value of σq at εF ≫ h̄ω, while the intraband conductivity continues to grow with increasing εF.
Both the interband and intraband c-c conductivities are actually functions of two dimensionless combinations h̄ω/kBT and εF/kBT. Once the temperature is reduced, Reσintra drops as well due to reduced phase space for collisions near Fermi-surface. The dependence Reσinter(ω) becomes more abrupt in the vicinity of interband threshold h̄ω = 2εF. Thereby, cooling of graphene sample is advantageous for achieving the negative dynamic conductivity. This is confirmed by Fig. 4. In Figs. 4(a) and 4(b), we show the real part of the net dynamic conductivity versus frequency ω/2π and quasi-Fermi energy εF (values of Reσ are marked by colors) at the room temperature T = 300 K (A) and at T = 200 K (B). It is worth noting that the temperature T in the above equations is in fact the effective temperature of the electron-hole system. Therefore, the reduction of this temperature can be achieved not only by decreasing of the ambient temperature, but also by direct cooling of this system at a certain pumping conditions. In particular, the cooling of electron-hole system in the optically and electrically pumped graphene lasing structures due to the emission of optical phonons can be substantial [5, 27].
In Fig. 5(a) we show the threshold lines of the negative net dynamic conductivity [which are solutions of the equation Reσ(ω, εF)= 0] at different values of background dielectric constant. The normalized threshold frequency h̄ω0/kBT slowly moves to lower values as κ0 increases. This is illustrated in Fig. 5(b). At room temperature and κ0 = 1, the negative conductivity could be attained only at frequencies ω/2π ≥ 6.5 THz. For graphene on SiO2 substrate [κ0 = (κSiO2 + 1)/2 = 2.5], the room temperature threshold lies at ω0/2π ≈ 6 THz. As discussed before, such slow decrease in threshold frequency is owing to the Thomas-Fermi screening of Coulomb interaction. Note that screening can be further enhanced by placing graphene close to the metal gate. In gated graphene, the Fourier component of Coulomb scattering potential VC(q) gains an additional factor [1− exp(−2qd)], where d is the distance to the gate.
The dependences of the inter- and intraband (carrier-carrier) conductivities on the quasi-Fermi energy at fixed frequency are shown in Fig. 6. The interband conductivity reaches its ultimate value of σq at εF ≫ h̄ω/2. Intuitively, Reσintra and the corresponding radiation absorption coefficient should be proportional to the number of particle-particle collisions in unit time. This number is proportional to the carrier density squared, at least for classical carrier systems. In reality, the dependence of Reσintra on the quasi-Fermi energy εF appears to be almost linear. This, first of all, results from the screening and also from the restricted phase space for collisions in Fermi-systems in a narrow layer near the Fermi-surface.
4. Discussion of the results
The obtained relation for the intraband conductivity associated with the c-c scattering [Eq. (13)] corresponds to a Drude-like dependence Reσintra ∝ ω−2. The latter tends to infinity at small frequencies. This tendency should be cut off at ω ∼ ν, where ν is the electron collision frequency. At such small frequencies, the finite lifetime of quasiparticles leads to inapplicability of the Fermi golden rule. However, one can estimate the carrier-carrier collision frequency and the corresponding low-frequency conductivity by solving the kinetic equation for the massless electrons in graphene. We have generalized the variational approach to the solution of kinetic equation  developed in [16,17] for intrinsic graphene, to the case of symmetrically pumped graphene. As a result, the following low-frequency dependence of intraband conductivity due to c-c scattering was obtained:
At intermediate frequencies νcc ≪ ω ≤ kBT/h̄, the results obtained from the kinetic equation[Eq. (16)] and the Fermi golden rule [Eq. (13)] coincide. At room temperature, zero Fermi energy, and background dielectric constant κ0 = 1, the characteristic collision frequency is equal to νcc ≈ 1013 s−1. Once κ0 or εF are increased, the collision frequency decreases due to screening of the Coulomb potential. Hence, for the ’target’ frequencies ω/(2π)= 5 − 10 THz one can readily use the golden rule result [Eq. (13)] to evaluate the carrier-carrier scattering contribution to the optical conductivity.
Until now we considered the intraband conductivity due to the c-c collisions only, as if other scattering processes were removed. In realty, the acoustic phonon contribution is unavoidable as well. The corresponding collision frequency can be presented as , where ν0 ≃ 3×1011 s−1 at room temperature . Thus, at least for not very large values of κ0, the c-c scattering contribution to intraband conductivity appears to be larger than the acoustic phonon contribution. The momentum relaxation time due to scattering by randomly distributed charged impurities with average density ni is
In our estimates for the Coulomb scattering probability, we have used the static (Thomas-Fermi) approximation for dielectric function of graphene. The use of the dynamic dielectric function can be mandatory to avoid the divergence in collision integrals for collinear scattering . However, in Eqs. (10) and (12) such a divergence does not appear due to the presence of factor Δn2. The difference between Reσintra obtained using static and dynamic screening is below 10 %.
As known, the e-e and h-h scattering does not lead to the finite resistivity and the absorption of radiation by the carriers with the parabolic dispersion law (in the absence of umklapp processes ). In graphene, such an effect is possible due to the momentum-velocity decoupling. Formally, e-e collisions in pumped graphene are of the same “strength” as e-h collisions [see Eqs. (9) and (11)]. However, the numerical comparison of Eqs. (9) and (11) shows that the e-e and h-h collisions play a minor role in the radiation absorption in the pumped graphene. The major contribution to Reσcc comes from the electron-hole collisions, while Reσee actually constitutes less than 10% of Reσcc (see Fig. 7(a). This can be understood by analyzing the change in current in a single collision act. The corresponding change is given by the terms (Δnee)2 and (Δneh)2 in Eqs. (10) and (12). Expanding in powers of q and averaging over directions, one obtains and . The latter implies that collisions of electrons with sufficiently different energies only play significant role in the e-e and h-h scattering-assisted absorption. For electron-hole collisions there is no such a restriction.
For the electron-hole system in equilibrium, the relative contributions of e-e and e–h scattering to the intraband conductivity are much different [see Fig. 7(b)]. The h-h and e–h contributions drop quickly with increasing the Fermi energy due to vanishingly small number of holes. The remaining part of intraband conductivity due to the e-e collisions is of order of 0.1σq at ω/(2π)= 6 THz and κ0 = 1. The corresponding contribution to radiation absorption can be in principle detected for rather clean samples.
Recent experiments have shown that the THz radiation gain in the pumped graphene can greatly exceed 2.3 % . This phenomenon is attributed to the self-excitation of surface plasmon-polaritons in electron-hole system with population inversion . The threshold of plasmon self-excitation in the pumped graphene is also governed by the ratio of the negative interband and positive intraband conductivities. The latter results mainly from the e-h collisions . The presented calculations show that terahertz plasmons are rather amplified than damped in the pumped graphene, at least at high pumping levels and large background dielectric constants.
We have shown that in graphene there exists a strong mechanism of the intraband Drude-like radiation absorption, assisted by the carrier-carrier collisions. In such a process, a carrier absorbs a photon and transfers the excess energy and momentum to the other carrier, electron or hole. The radiation absorption assisted by the electron-electron and hole-hole collisions is possible only in semiconductors with non-parabolic bands, and graphene (with its linear energy spectrum of electrons and holes) is one of brightest representatives of such a family. We have evaluated the carrier-carrier scattering contribution to the intraband conductivity using the second-order perturbation theory and the Fermi golden rule. As demonstrated, the radiation absorption due to the carrier-carrier collisions can markedly surpass the Drude absorption due to the scattering on thermal or static disorder. This is especially pronounced in the pumped graphene with a large number of carriers in both bands.
We have studied the effect of carrier-carrier scattering on the conditions of the negative dynamic conductivity in pumped graphene by comparing the positive intraband dynamic conductivity due to the carrier-carrier collisions and the negative interband conductivity. We have shown that at room temperature, the net negative dynamic conductivity is attainable at frequencies above ∼ 6.5 THz in suspended graphene. This frequency threshold can be shifted to lower values (∼ 2.5 THz at room temperature) for graphene on high-κ substrates, as the presence of dielectric partially screens the Coulomb interaction. Cooling of the electron-hole system also reduces the negative conductivity threshold. Our results show that screening of Coulomb interaction by carriers in graphene itself (Thomas-Fermi screening) significantly affects the real part of intraband conductivity. Due to the screening, the c-c contribution to the intraband conductivity grows slowly (almost linearly) with increasing quasi-Fermi energy of nonequilibrium electrons and holes.
Appendix evaluation of Coulomb integrals
We introduce the dimensionless energies of incident particles ε1 = k1+ and ε2 = k2−, and the transferred energy δε = k1+ − k1− − w/2 = k2+ − k2− − w/2. Here, for brevity, w = h̄ω/(kBT). The kernel of the e-e Coulomb integral depends only on those energies and the transferred momentum Q:Eq. (19) can be rewritten as Eq. (22), one finds the limits of integration over energies in Eq. (20) from general requirements cosh ui > 1, |cosvi| < 1, i = 1,2: Eq. (24) k1+k1−k2+k2− is the Jacobian determinant for elliptic coordinates.
The envelope function overlap factors are also concisely expressed in the elliptic coordinates:Eq. (26) contains the denominator divergent at w → 0, which manifests the collinear scattering anomaly for massless particles [9, 16, 17]. This divergence is removed by the similar term in the expression for , which stands in the numerator.
The authors are grateful to M. Ryzhii for useful comments. The work at RIEC was supported by the Japan Society for Promotion of Science (Grant-in-Aid for Specially Promoting Research # 23000008), Japan. The work at IPT RAS was supported by the Russian Foundation of Basic Research (grant # 14-07-00937). The work by D.S was also supported by the grant of the Russian Foundation of Basic Research # 14-07-31315 and by the JSPS Postdoctoral Fellowship For Foreign Researchers (Short-term), Japan.
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