Induced nonlinear cross sections of conductive electrons scattering on the charged impurities in doped graphene
A. K. Avetissian, A. G. Ghazaryan, Kh. V. Sedrakian, B. R. Avchyan
aa r X i v : . [ phy s i c s . op ti c s ] O c t Induced nonlinear cross sections of conductive electrons scattering on the chargedimpurities in doped graphene
A.K. Avetissian, A.G. Ghazaryan, ∗ Kh.V. Sedrakian, and B.R. Avchyan
Centre of Strong Fields, Yerevan State University, 1 A. Manukian, Yerevan 0025, Armenia (Dated: September 9, 2018)Relativistic quantum theory of induced scattering of 2D Dirac particles by electrostatic field ofimpurity ion (in the Born approximation) in the doped graphene at the presence of an externalelectromagnetic radiation field (actually terahertz radiation, to exclude the valence electrons exci-tations at high Fermi energies) has been developed. It is shown that the strong coupling of masslessquasiparticles in the quantum nanostructures to a strong electromagnetic radiation field leads to thestrongly nonlinear response of graphene, which opens diverse ways for manipulating the electronictransport properties of conductive electrons by coherent radiation fields.
PACS numbers: 42.50.Hz, 34.80.Qb, 32.80.Wr, 31.15.-p
I. INTRODUCTION
The physics of graphene [1, 2] with the current ad-vanced technologies opens large research and appliedfields including wide spectrum of investigations from low-energy condensed matter physics to quantum electro-dynamic (QED) effects [3, 4] due to exotic features ofquasiparticles in graphene that behave like massless ”rel-ativistic” Dirac fermions (with the Fermi velocity muchless than the light speed in vacuum: v F = 10 cm / s)and obey a two-dimensional (2D) Dirac equation [5, 6].These properties of quasiparticles lead to various appli-cations of graphene to replicate the fundamental nonlin-ear QED processes in much weaker electromagnetic (EM)fields compared to common materials and vacuum wheresuperintense laser fields of ultrarelativistic intensities arerequired for observation of nonlinear phenomena such asproduction of electron-positron pairs from the vacuum,nonlinear Compton scattering, multiphoton stimulatedbremsstrahlung (SB) etc, proceeding actually in the cur-rent strong and superstrong laser fields [7–10].Among the important processes induced by externalradiation fields SB is one of the first stimulated effects atlaser-matter interaction revealed immediately after theinvention of lasers [11]. The latter is basic mechanism ofenergy exchange between the charged particles and planemonochromatic wave in plasma-like media to provide theenergy-momentum conservation law for real absorption-emission processes. What concerns the electrons elasticscattering on impurity ions in graphene, there are manypapers with consideration of this basic scattering effectwhich have been described mainly within the frameworkof perturbation theory by electrostatic potential (see,e.g., [12–18]). Regarding the SB process in graphene atmoderate intensities of stimulated radiation, in case of itslinear absorption by electrons (or holes), at the presenttime there are extensive investigations carried out in the ∗ Electronic address: [email protected] scope of the linear theory, see, e.g. [19–23]. The linearabsorbance of single layer graphene from infrared up tovisible spectral range is πα ≈ .
3% which depends solelyon the fine-structure constant α = 1 /
137 [24]. Conse-quently, taking into account the graphene thickness, theabsorption coefficient will be 10 cm − , which is a remark-ably high value for absorption effect. On the other hand,high EM radiation absorption by ultrasmall volumes isa very challenging property for shielding materials usedin nanoelectronics, aviation, and space industry, wherestrict requirements on lightness and smallness of mate-rials exist. In this aspect, due to its unique propertiesgraphene can act as a THz emitting device and/or shield-ing material for nanodevices, enabling to bridge so-calledTHz gap [25, 26]. Meanwhile, consideration of multipho-ton SB process in graphene in moderately strong laserfields as a basic mechanism for radiation absorption to-wards the mentioned applications is absent up to now.The latter is given in the current paper.In general, the interaction of a free electron with theEM wave is described by the dimensionless relativisticinvariant parameter of intensity ξ = eE λ/ (2 πmc ) [7],which represents the wave electric field (with amplitude E ) work on a wavelength λ in the units of electron restenergy. Particularly for THz photons ~ ω ∼ .
01 eV, mul-tiphoton effects take place at ξ ∼ I ξ ∼ Wcm − , while the massless electron-wave interaction in graphene is characterized by the di-mensionless parameter χ = ev F E / (cid:0) ~ ω (cid:1) [8], which rep-resents the work of the wave electric field on a period 1 /ω in the units of photon energy ~ ω . Depending on the valueof this parameter χ , three regimes of the wave-particle in-teraction may be established: χ ≪ χ ≫ χ > I χ = χ × . × Wcm − [ ~ ω/ eV] . Comparison ofthis intensity threshold with the analogous one for thefree electrons or situation in common atoms shows theessential difference between the values of these thresh-olds: I ξ /I χ ∼ . Thus, for realization of multiphotonSB in graphene one can expect 10 times smaller inten-sities than for SB in atoms [9], [10], [11, 31, 32].In the present work, the influence of multiphoton ef-fects in SB absorption process with moderately stronglaser fields is considered. Here the selected frequencyrange of terahertz radiation excludes the valence elec-trons excitations at high Fermi energies.In Sec. I the scattering rates and total multiphotoncross-sections for SB of conduction electrons in graphenehave been obtained. The analytic formulas in case ofscreened Coulomb potential have been analyzed numer-ically in Sec. II. We have also present results for theangular dependence of scattered Fermi electrons on thelaser radiation intensities. Conclusions are given in Sec.III. II. MULTIPHOTON AMPLITUDES ANDCROSS-SECTIONS OF SB IN GRAPHENE
Below we will develop the relativistic scattering theoryfor the 2D Dirac fermions on arbitrary electrostatic po-tential U ( r ) of an impurity ion in doped graphene andinteracting with an external EM wave field of moderateintensities. To exclude the valence electrons excitationsat high Fermi energies in graphene, we will assume for aEM wave actually a terahertz radiation.Let us determine the scattering Green function formal-ism in the Born approximation by potential U ( r ). Notethat the first nonrelativistic treatment of SB in the Bornapproximation has been carried out analytically in thework [11], and then this approach has been extended tothe relativistic domain [31].Transition amplitude of SB process in the EM wavefield from the state with the canonical momentum p ( p x , p y ) to the state with momentum p ( p x , p y ) ingraphene plane ( x, y ) can be written as: C pp = − i ~ Z Ψ + p ( r , t ) U ( r )Ψ p ( r , t ) dtd r , (1)where bispinor function Ψ + is the complex conjugationof Ψ.The fermion particle wave function Ψ p ( r , t ) in thestrong EM wave field may be presented in the form:Ψ p ( r , t ) = exp (cid:18) i ~ pr (cid:19) f p ( t ) , (2)with the spinor wave function f p determined as follows: f p ( t ) = 1 √ S (cid:18) e i Θ( p + ec A ( t )) (cid:19) e − i Ω( p ,t ) , (3)where the instantaneous temporal phase Ω( p , t ) is de-fined as: Ω( p , t ) = v F ~ R q(cid:0) p x + ec A x (cid:1) + p y dt , the func-tion Θ( p + ec A ( t )) is the angle between the vectorsof particle kinematic momentum in the EM field p = p (cid:0) p x + ec A x , p y (cid:1) and the wave vector potential A ( t ) = − c R t E ( t ′ ) dt ′ (unit vector b e is directed along the axis OX ), and parameter S is the quantization area -graphenelayer surface area. In terms of these parameters, thegraphene linear dispersion law for quasiparticles energy-momentum E ( p ) defined by the characteristic Fermi ve-locity v F , reads: E = ± v F | p | = ± v F q p x + p y , wherethe upper sign corresponds to electrons and the lowersign -to holes.As is known, the state of an electron in the field of astrong EM wave, and consequently, the cross section ofSB essentially depends on the wave polarization. Hencewe will consider the case of certain polarization of thewave, let a linear.Let us first study a single electron scattering oncharged impurity in graphene and interacting simulta-neously with an EM radiation field E ( t ) (it is clear thatat such small Fermi velocities of scattered particles theplane monochromatic wave field of frequency ω will turninto the uniform periodic electric field of frequency ω : E ( t ) = E cos ωt ) let polarized along the OX axis.For determination of spinor wave function f p we willuse the results of the paper [33] with the obtained formulafor transition amplitude: C pp = − i ~ Z f + p ( t ) U ( r ) f p ( t ) exp (cid:18) i ~ ( p − p ) r (cid:19) dtd r , (4)or represented in the form: C pp = − i ~ Z f + p ( t ) e U (cid:18) p − p ~ (cid:19) f p ( t ) dt, (5)i.e., the transition amplitude C pp (5) is depended by theFourier transform of the scattering potential: e U (cid:18) p − p ~ (cid:19) = Z exp (cid:18) i ~ ( p − p ) r (cid:19) U ( r ) d r . (6)For the impurity potential of the arbitrary form e U h p − p ~ i from the relation (5) we have: C pp = − i ~ S e U (cid:20) p − p ~ (cid:21) × Z (cid:16) e i [ Θ( p + ec A ( τ )) − Θ( p + ec A ( t )) ] (cid:17) (7) e − i ~ v F R τ (cid:20)q ( p x + ec A x ) + p y − q ( p x + ec A x ) + p y (cid:21) dt dτ . In accordance to the transition amplitude (7), impuritypotential can be expressed in the following form: C pp = Z B ( τ ) e − i ~ v F ( P − P ) τ dτ , (8)where B ( τ ) = − i ~ S e U (cid:20) p − p ~ (cid:21) × (cid:16) e i [ Θ( p + ec A ( t )) − Θ( p + ec A ( t )) ] (cid:17) × e − i ~ v F R τ (cid:20)(cid:18)q ( p x + ec A x ) + p y − P (cid:19) − (cid:18)q ( p x + ec A x ) + p y − P (cid:19)(cid:21) dt (9)is the periodic function of time, and the time-dependedmodules of the ”quasimomentums” P , P are defined as: P = ω π Z πω r(cid:16) p x + ec A x ( t ) (cid:17) + p y dt,P = ω π Z πω r(cid:16) p x + ec A x ( t ) (cid:17) + p y dt. Here making a Fourier transformation of the function B ( t ) over t , using the known relations B ( t ) = ∞ X n = −∞ e B n exp( − inωt ) , (10) e B n = ω π Z π/ω B ( t ) exp( inωt ) dt, (11)and carrying out the integration over t in the formula (8),we obtain: C ( n ) pp = 2 π ~ e B n δ (v F P − v F P − n ~ ω ) . (12)Within the Born approximation, the differential prob-ability W pp of SB per unit time, from the electron orhole state with two-dimensional momentum p to a statewith momentum p in the phase space Sd P / (2 π ~ ) is de-scribed by the formula: W pp = lim t →∞ t (cid:12)(cid:12) C pp (cid:12)(cid:12) P dP dθ S (2 π ~ ) , (13)where dθ is the differential scattering angle (linear).Dividing the differential probability W pp (12) of SBby initial flux density v F and integrating over dP , we ob-tain the differential cross-section of SB for quasiparticlesin doped graphene: d Λ dθ = ∞ X n = − n m d Λ ( n ) dθ , (14)where d Λ ( n ) dθ = | P | v F (cid:12)(cid:12)(cid:12) e B n (cid:12)(cid:12)(cid:12) (cid:12)(cid:12)(cid:12)(cid:12) v F P =v F P + n ~ ω (15)is the partial differential cross-section of n -photon SBwith maximum number of emitted photons n m . The to-tal scattering cross-section d Λ /dθ will be obtained mak-ing summation over the photon numbers in the formulafor differential partial cross-sections d Λ ( n ) /dθ (14). Thelatter may be represented in the form: d Λ ( n ) dθ = (cid:12)(cid:12)(cid:12)(cid:12) e U (cid:20) p − p ~ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T d (cid:18) tT (cid:19) (cid:16) e i [ Θ( p + ec A ( t )) − Θ( p + ec A ( t )) ] (cid:17) exp( inωt ) × e − i ~ v F R t (cid:20)(cid:18)q ( p x + ec A x ) + p y − P (cid:19) − (cid:18)q ( p x + ec A x ) + p y − P (cid:19)(cid:21) dt ′ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) × δ (cid:18) P − P − n ~ ω v F (cid:19) P dP π ~ )v F ~ . (16)For the comparison with the known results in particularcases of scattering, let we obtain the partial differentialcross-sections of SB in the case of n = ±
1. We willproduce the expansion in Eq. (16) into a Taylor seriesand keep only the terms of the first order over the electricfield. Then we can obtain an asymptotic formula forthe partial differential cross-sections of SB process in theweak wave field (linear theory): d Λ ( ± dθ = χ π v F ~ (cid:12)(cid:12)(cid:12)(cid:12) e U (cid:20) p − p ~ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) × (cid:12)(cid:12)(cid:12) ( p x − p x ) (cid:16) e i [Θ( p ) − Θ( p )] (cid:17) ∓ i ~ ω v F (cid:18) p y p − p y p (cid:19) e i [Θ( p ) − Θ( p )] (cid:12)(cid:12)(cid:12)(cid:12) × δ (cid:18) P − P ∓ ~ ω v F (cid:19) P dP. (17)Comparing the n -photon cross-section d Λ ( n ) (16) of SBprocess with the elastic one, we conclude that formula(16) at A ( τ ) = 0 ( n = 0) passes to elastic scatteringcross-section d Λ elast [16], which is the analog of the Mottformula in 2D scattering theory: d Λ elast dθ = | p | π v F ~ (cid:12)(cid:12)(cid:12)(cid:12) e U (cid:20) p − p ~ (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) × (cid:12)(cid:12)(cid:12)(cid:16) e i [Θ( p ) − Θ( p )] (cid:17)(cid:12)(cid:12)(cid:12) . (18)The phase term (cid:0) i (cid:2) Θ( p + ec A ( τ )) − Θ( p + ec A ( τ )) (cid:3)(cid:1) in Eq. (16) at A ( τ ) = 0 is the overlap factor (cid:0) e − iθ q (cid:1) (cid:0) e iθ q (cid:1) = 2 (1 + cos θ q ) , (19)where θ q = Θ( p ) − Θ( p ). The term (19) known asa Berry phase term arising from the inherent sublatticesymmetry, which with a graphene fourfold ground statedegeneracy arising from the spin and valley factors, re-stricts the carriers from backscattering. III. DIFFERENTIAL CROSS-SECTIONS OF SBON THE SCREENED COULOMB POTENTIALOF IMPURITY IONS IN GRAPHENE
Now we utilize Eq. (16) in order to obtain the differ-ential cross-section in particular case of SB process on ascreened Coulomb potential of impurity ions in graphene[15], [18], [34–37]. In accordance with [15], the Fouriertransform e U ( q ) = R U ( r ) e − i qr d r of a charged impuritycenter potential can be written as: e U ( q ) = 2 πe e κqǫ ( q ) , (20)where ǫ ( q ) ( q = | q | ) is the 2D finite temperature staticdielectric (screening) function in random phase approx-imation (RPA) appropriate for graphene [37], given bythe formula ǫ ( q ) = 1 + q s q × ( − πq k F , q ≤ k F − √ q − k F q − q sin − k F /q k F , q > k F . (21)Here k F = ε F / ~ v F is 2D Fermi wave vector, q s =4 e k F / ( ~ e κ v F ) is the effective graphene 2D Thomas-Fermi wave vector, and e κ = κ (1 + πr s /
2) is the ef-fective dielectric constant of a substrate. The ratio ofthe potential to the kinetic energy in an interactingquantum Coulomb system is measured by the dimen-sionless Wigner-Seitz radius r s = e /κ ~ v F , where κ isthe background lattice dielectric constant of the system, e / ~ v F ≃ .
18 is “effective fine-structure constant” ingraphene (in the vacuum). Since we are interested inactual laser pulses for external EM radiation, at the con-sideration of numerical results it is convenient to repre-sent the differential cross-sections of SB on the chargedimpurities in the form of dimensionless quantities.Taking into account Eqs. (7), (16), and (20), we obtainthe following form for the dimensionless partial differen-tial cross-sections of SB process d Λ ( n ) /dθ in the field oflinearly polarized EM wave with the dimensionless vector (a) d Λ (0) /d θ [A o ]-3 -2 -1 0 1 2 3 θ - θ [rad] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 θ [ r ad ] Λ (0) /d θ [A o ]-3 -2 -1 0 1 2 3 θ - θ [rad] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 θ [ r ad ] Λ (0) /d θ [A o ]-3 -2 -1 0 1 2 3 θ - θ [rad] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 θ [ r ad ] Λ (0) /d θ [A o ]-3 -2 -1 0 1 2 3 θ - θ [rad] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 θ [ r ad ] FIG. 1: (Color online) Partial differential cross section d Λ ( n ) /dθ (in angstrom) of SB process for photon number n = 0 vs the electron deflection angle θ − θ and incidentangle θ for linear polarization of EM wave of intensities: (a) χ = 0, (b) χ = 1, (c) χ = 5, and (d) χ = 7. (a) d Λ (1) /d θ [A o ]-3 -2 -1 0 1 2 3 θ - θ [rad] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 θ [ r ad ] Λ (1) /d θ [A o ]-3 -2 -1 0 1 2 3 θ - θ [rad] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 θ [ r ad ] Λ (1) /d θ [A o ]-3 -2 -1 0 1 2 3 θ - θ [rad] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 θ [ r ad ] Λ (1) /d θ [A o ]-3 -2 -1 0 1 2 3 θ - θ [rad] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 θ [ r ad ] FIG. 2: (Color online) Same as Fig. 1 but for photon number n = 1 and laser intensities: (a) χ = 1, (b) χ = 3, (c) χ = 5,and (d) χ = 7. potential A ( t ) = − b e χ sin(2 πτ ) (unit vector b e is directedalong the axis OX ) : d Λ ( n ) λdθ = 1300 (cid:18) r s πr s (cid:19) | p − p | ǫ ( | p − p | ) × (cid:12)(cid:12)(cid:12)(cid:12)Z dτ (cid:16) e i [Θ( p − b e χ sin(2 πτ )) − Θ( p − b e χ sin(2 πτ ))] (cid:17) × exp (cid:26) i πnτ − πi Z τ (cid:20)(cid:18)q ( p x − χ sin(2 πτ ′ )) + p y − P (cid:19) − (cid:18)q ( p x − χ sin(2 πτ ′ )) + p y − P (cid:19)(cid:21) dτ ′ (cid:27)(cid:12)(cid:12)(cid:12)(cid:12) × δ (cid:0) P − P − n (cid:1) P dP . (22)In Eq. (22) the dimensionless momentum, energy, time,and relativistic invariant intensity parameter of EM waveintroduced as follows: p x,y = v F ~ ω p x,y , E = E ~ ω ,k F = E F ~ ω , dτ = dtT , χ = e v F ~ ω E . For numerical analysis of SB cross sections in graphenewe assume Fermi energy ε F = 20 ~ ω ( ε F ≫ n ~ ω ), co-herent EM radiation with energy of photons ~ ω = 0 . λ = 1 . × ˚A), dielectric environment constant κ = 2 . substrate [37], Wigner-Seitz radius r s = 0 . T = 0 . ε F .In the Figs. 1-5, the envelopes of partial differentialcross sections d Λ ( n ) /dθ (22) of SB in graphene for a lin-early polarized EM wave are shown as a function of theangle θ between the vectors of the electron initial mo-mentum and electric field strengths of a EM wave, θ − θ is the electron deflection angle. These figures illustrateSB at different intensities of stimulated wave. Thus, Fig.1 illustrates elastic part of bremsstrahlung ( n = 0), whileFigs. 2-5 – SB for different number of absorbed photons(or emitted photons at n < n = 1), Fig. 3 – two-photon SB ( n = 2), Fig. 4 – three-photon SB ( n = 3), and Fig. 5 – four-photon SB ( n = 4),respectively. The angular dependences of partial differ-ential cross sections in these figures are displayed for in-tensities: in Fig. 1 (a) at χ = 0, (b) χ = 1 ( I χ = 3 × Wcm − ), (c) χ = 5 ( I χ = 7 . × Wcm − ), and (d) χ = 7 ( I χ = 1 . × Wcm − ), and in Figs. 2-5: (a) χ = 1, (b) χ = 3 ( I χ = 2 . × Wcm − ), (c) χ = 5,and (d) χ = 7, respectively. As one can see, the an-gular distribution becomes more asymmetrical with theincreasing of the wave intensity. The maximum valuesof the cross sections correspond to different values of thedeflection angle θ − θ .In Fig. 6 we plot the dependence of envelopes of partialcross sections n Λ ( n ) for a linearly polarized wave uponthe number of emitted or absorbed photons. The en-velopes are obtained via integrating of the partial dif-ferential cross section of SB process d Λ ( n ) /dθ (22) overscattering angle of the outgoing electron for diverse laserintensities. The angle θ between the initial electron mo-mentum and wave electric field is taken to be: 0, π/ π/ (a) 10·d Λ (2) /d θ [A o ]-3 -2 -1 0 1 2 3 θ - θ [rad] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 θ [ r ad ] Λ (2) /d θ [A o ]-3 -2 -1 0 1 2 3 θ - θ [rad] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 θ [ r ad ] Λ (2) /d θ [A o ]-3 -2 -1 0 1 2 3 θ - θ [rad] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 θ [ r ad ] Λ (2) /d θ [A o ]-3 -2 -1 0 1 2 3 θ - θ [rad] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 θ [ r ad ] FIG. 3: (Color online) Same as Fig. 2 but for photon number n = 2. (a) 10 ·d Λ (3) /d θ [A o ]-3 -2 -1 0 1 2 3 θ - θ [rad] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 θ [ r ad ] Λ (3) /d θ [A o ]-3 -2 -1 0 1 2 3 θ - θ [rad] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 θ [ r ad ] Λ (3) /d θ [A o ]-3 -2 -1 0 1 2 3 θ - θ [rad] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 θ [ r ad ] Λ (3) /d θ [A o ]-3 -2 -1 0 1 2 3 θ - θ [rad] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 θ [ r ad ] FIG. 4: (Color online) Same as Fig. 2 but for photon number n = 3. intensities I χ ∼ Wcm − . Thus, for these intensi-ties multiphoton SB process opens new channels for thewave absorption, and we can expect strong deviation ofabsorbance of a single layer doped graphene from linearone, which for frequencies smaller than Fermi energy iszero [24].In Fig. 7 we display laser-modified elastic cross sectionΛ (0) versus intensity parameter for several initial angle θ between the electron momentum and wave electric field.As is seen from this figure, in the presence of strong ra-diation field elastic cross section is essentially modifiedand decreases with the increase of induced radiation in-tensity. The latter opens up possibility for manipulatingof electronic transport properties of the doped grapheneby coherent radiation field. (a) 10 ·d Λ (4) /d θ [A o ]-3 -2 -1 0 1 2 3 θ - θ [rad] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 θ [ r ad ] Λ (4) /d θ [A o ]-3 -2 -1 0 1 2 3 θ - θ [rad] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 θ [ r ad ] Λ (4) /d θ [A o ]-3 -2 -1 0 1 2 3 θ - θ [rad] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 θ [ r ad ] Λ (4) /d θ [A o ]-3 -2 -1 0 1 2 3 θ - θ [rad] 0 0.2 0.4 0.6 0.8 1 1.2 1.4 θ [ r ad ] FIG. 5: (Color online) Same as Fig. 2 but for photon number n = 4. -2-1 0 1 2 3 4 (a) n Λ ( n ) [ A o ] χ =1 χ =3 χ =5 χ =7-2-1 0 1 2 3 (b) n Λ ( n ) [ A o ] χ =1 χ =3 χ =5 χ =7-8-4 0 4 8 -9 -7 -5 -3 -1 1 3 5 7 9(c) n Λ ( n ) [ A o ] n χ =1 χ =3 χ =5 χ =7 FIG. 6: (Color online) Envelopes of partial absorption-emission cross sections n Λ ( n ) (in angstrom) as a function ofthe number of emitted or absorbed photons. The angle θ be-tween the initial electron momentum and wave electric fieldis taken: (a) 0, (b) π/ π/ IV. CONCLUSION
We have presented the theoretical treatment of themultiphoton stimulated bremsstrahlung process in dopedgraphene. On the base of the ”relativistic” quantum theory it has been investigated the induced scatteringof 2D Dirac particles on the charged impurity ions ofarbitrary electrostatic potential in the Born approxima-tion and in an external electromagnetic radiation field(actually terahertz radiation to exclude the valence elec-trons excitations at high Fermi energies). The obtainedrelativistic analytical formulas for SB at the linear po- Λ ( ) [ A o ] χ θ =0 θ = π/6θ = π/2 FIG. 7: Elastic cross section Λ (0) versus intensity parameterfor several initial angle θ between the electron momentumand wave electric field. larization of EM wave have been analyzed numericallyfor screened Coulomb potential. The latter shows thatSB in graphene in the presence of strong radiation fieldis essentially nonlinear, and the multiphoton absorp-tion/emission processes play significant role already atmoderate laser intensities. For these intensities multi-photon SB process opens new channels for the wave ab-sorption, which will essentially modify absorbance of sin-gle layer doped graphene compared to the case of linearabsorbance (this problem will be presented in the comingpaper). Besides, the laser-modified elastic cross section issubstantially modified and decreases with the increase ofradiation field intensity. The latter opens up possibilityfor manipulating of electronic transport properties of thedoped graphene by coherent radiation field. Acknowledgments
The authors are deeply grateful to the Prof. Hamlet K.Avetissian and Dr. G.F. Mkrtchian for permanent dis-cussions during the work on the present paper, for valu-able comments and recommendations. This work wassupported by the State Committee of Science MES RA,in the frame of the research project No. 15T-1C013. [1] K. S. Novoselov et al., “Electric field effect in atomicallythin carbon films”, Science (5696), 666–669 (2004),http://dx.doi.org/10.1126/science.1102896. [2] A. H. Castro Neto et al., “The electronic propertiesof graphene”, Rev. Mod. Phys. (1), 109–162 (2009),http://dx.doi.org/10.1103/RevModPhys.81.109. [3] M. I. Katsnelson, K. S. Novoselov, and A. K.Geim, “Chiral tunnelling and the Klein paradoxin graphene”, Nature Phys. , 620–625 (2006),http://dx.doi.org/10.1038/nphys384.[4] M. I. Katsnelson and K. S. Novoselov, “Graphene: newbridge between condensed matter physics and quantumelectrodynamics”, Solid State Commun. (1–2), 3–13(2007), http:// dx.doi.org/10.1016/j.ssc.2007.02.043.[5] A. K. Geim, “Graphene: Status andprospects,” Science (5934), 1530–1534 (2009),http://dx.doi.org/10.1126/science.1158877.[6] K. S. Novoselov et al., “Two-dimensional gas of mass-less Dirac fermions in graphene”, Nature , 197–200(2005), http://dx.doi.org/10.1038/nature04233.[7] H. K. Avetissian, ”Relativistic Nonlinear Electrodynam-ics”, The QED Vacuum and Matter in Super-Strong Ra-diation Fields, Springer, the Netherlands, 2016.[8] H. K. Avetissian et al., “Creation of particle-hole super-position states in graphene at multiphoton resonant ex-citation by laser radiation,” Phys. Rev. B (11), 115443(2012), http://dx.doi.org/10.1103/PhysRevB.85.115443.[9] H. K. Avetissian, A. G. Ghazaryan, G. F.Mkrtchian, “Relativistic theory of inverse-bremsstrahlung absorption of ultrastrong laser ra-diation in plasma”, J. Phys. B , 205701 (2013),http://dx.doi.org/10.1088/0953-4075/46/20/205701[10] H. K. Avetissian, A. G. Ghazaryan, H. H. Matevosyan,G. F. Mkrtchian, “Microscopic nonlinear relativisticquantum theory of absorption of powerful x-ray radi-ation in plasma”, Phys. Rev. E , 043103 (2015),http://dx.doi.org/10.1103/PhysRevE.92.043103.[11] F. V. Bunkin, A. E. Kazakov, M. V. Fedorov, “Inter-action of intense optical radiation with free electrons(nonrelativistic case)”, Sov. Phys.-Usp. , 416 (1973),http://iopscience.iop.org/0038-5670/15/4/R04; M. H.Mittleman, “Introduction to the theory of laser-atom in-teractions”, Plenum, New York, 1993.[12] T. Ando, “Screening effect and impurity scattering inmonolayer graphene”, J. Phys. Soc. Jpn. , 074716(2006), http://dx.doi.org/10.1143/JPSJ.75.074716.[13] J.-H. Chen, C. Jang, S. Adam, M. S. Fuhrer, E. D.Williams, and D.M. Ishigami, “Charged-impurity scat-tering in graphene”, Nature Physics , 377 (2008),http://dx.doi.org/10.1038/nphys935.[14] K. Nomura and A. H. MacDonald, “Quan-tum Hall ferromagnetism in graphene”,Phys. Rev. Lett. , 256602 (2006),http://dx.doi.org/10.1103/PhysRevLett.96.256602.[15] E. H. Hwang, S. Adam, and S. Das Sarma,“Carrier transport in two-dimensional graphenelayers”, Phys. Rev. Lett. , 186806 (2007),http://dx.doi.org/10.1103/PhysRevLett.98.186806.[16] D. S. Novikov, “Elastic scattering theory and trans-port in graphene”, Phys. Rev. B , 245435 (2007),http://dx.doi.org/10.1103/PhysRevB.76.245435.[17] Y.-W. Tan, Y. Zhang, K. Bolotin, Y. Zhao, S. Adam,E.H. Hwang, S. Das Sarma, H. L. Stormer, and P. Kim,“Measurement of scattering rate and minimum conduc-tivity in graphene”, Phys. Rev. Lett. , 246803 (2007),http://dx.doi.org/10.1103/PhysRevLett.99.246803.[18] M. I. Katsnelson, “Nonlinear screening of charge impu-rities in graphene”, Phys. Rev. B. , 201401 R (2006),http://dx.doi.org/10.1103/PhysRevB.74.201401.[19] D. P. DiVincenzo and E. J. Mele, “Self-consistent effective-mass theory for intralayer screening in graphiteintercalation compounds”, Phys. Rev. B , 1685 (1984),http://dx.doi.org/10.1103/PhysRevB.29.1685.[20] N. H. Shon and T. Ando, “Quantum transport in two-dimensional graphite system”, J. Phys. Soc. Jpn. ,2421 (1998), http://dx.doi.org/10.1143/JPSJ.67.2421.[21] H. Suzuura and T. Ando, “Crossover from symplec-tic to orthogonal class in a two-dimensional honey-comb lattice”, Phys. Rev. Lett. , 266603 (2002),http://dx.doi.org/10.1103/PhysRevLett.89.266603.[22] N. M. R. Peres, F. Guinea, and A. H. Cas-tro Neto, “Electronic properties of disordered two-dimensional carbon”, Phys. Rev. B , 125411 (2006),http://dx.doi.org/10.1103/PhysRevB.73.125411.[23] S. Sun and J.-L. Zhu, “Impurity spec-tra of graphene under electric and magneticfields”, Phys. Rev. B , 155403 (2014),http://dx.doi.org/10.1103/PhysRevB.89.155403.[24] R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov,T. J. Booth, T. Stauber, N. M. R. Peres, and A.K. Geim, “Fine Structure Constant Defines VisualTransparency of Graphene”, Science , 1308 (2008),http://dx.doi.org/10.1126/science.1156965[25] J. Liang et al, “Electromagnetic interference shieldingof graphene/epoxy composites”, Carbon , 922 (2009),http://dx.doi.org/10.1016/j.carbon.2008.12.038.[26] T. Low and P. Avouris, “Graphene plasmonics for tera-hertz to mid-infrared applications”, ACS Nano , 1086(2014), http://dx.doi.org/10.1021/nn406627u.[27] E. G. Mishchenko, “Dynamic conductiv-ity in graphene beyond linear response”,Phys. Rev. Lett. (24), 246802 (2009),http://dx.doi.org/10.1103/PhysRevLett.103.246802.[28] P. N. Romanets and F. T. Vasko, “Rabi oscil-lations under ultrafast excitation of graphene”,Phys. Rev. B. (24), 241411(R) (2010),http://dx.doi.org/10.1103/PhysRevB.81.241411.[29] B. D´ora et al., “Rabi oscillations in Landau-quantizedgraphene”, Phys. Rev. Lett. , 036803 (2009),http://dx.doi.org/10.1103/PhysRevLett.102.036803.[30] B. D´ora and R. Moessner, “Nonlinear electric trans-port in graphene: quantum quench dynamics and theSchwinger mechanism”, Phys. Rev. B. (16), 165431(2010), http://dx.doi.org/10.1103/PhysRevB.81.165431.[31] M. M. Denisov, M. V. Fedorov, “Bremsstrahlung effecton relativistic electrons in a strong radiation field”, Sov.Phys. JETP , 779 (1968).[32] T. R. Hovhannisyan, A. G. Markossian, G. F.Mkrtchian, “On the theory of the relativistic cross-sections for stimulated bremsstrahlung on an arbi-trary electrostatic potential in the strong electro-magnetic field”, Eur. Phys. J. D , 17 (2002),http://dx.doi.org/10.1140/epjd/e2002-00110-7.[33] K. L. Ishikawa, “Nonlinear optical response of graphenein time domain”, Phys. Rev. B. (20), 201402(R)(2010), http://dx.doi.org/10.1103/PhysRevB.82.201402.[34] S. Adam, E. H. Hwang, V. M. Galitski,and S. Das Sarma, “A self-consistent theoryfor graphene transport”, PNAS , 18392,http://dx.doi.org/10.1073/pnas.0704772104.[35] T. Ando, A.B. Fowler , F. Stern, “Electronic propertiesof two-dimensional systems”, Rev. Mod. Phys. , 437(1982), http://dx.doi.org/10.1103/RevModPhys.54.437.[36] R. Saito, G. Dresselhaus, and M. S. Dresselhaus, “Phys- ical properties of carbon nanotubes”, Imperial CollegePress, London, UK, 1999.[37] E. H. Hwang and S. Das Sarma, “Dielectric func-tion, screening, and plasmons in two-dimensional graphene”, Phys. Rev. B.75