Ultrafast Carrier Recombination and Generation Rates for Plasmon Emission and Absorption in Graphene
Farhan Rana, Jared H. Strait, Haining Wang, Christina Manolatou
aa r X i v : . [ c ond - m a t . m e s - h a ll ] J un Ultrafast Carrier Recombination and Generation Rates for Plasmon Emission andAbsorption in Graphene
Farhan Rana, Jared H. Strait, Haining Wang, Christina Manolatou
School of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14853 ∗ Electron-hole generation and recombination rates for plasmon emission and absorption in grapheneare presented. The recombination times of carriers due to plasmon emission have been found tobe in the tens of femtoseconds to hundreds of picoseconds range. The recombination times dependsensitively on the carrier energy, carrier density, temperature, and the plasmon dispersion. Carriersnear the Dirac point are found to have much longer lifetimes compared to carriers at higher energies.Plasmons in a graphene layer on a polar substrate hybridize with the surface optical phonons andthis hybridization modifies the plasmon dispersion. We also present generation and recombinationrates of carriers due to plasmon emission and absorption in graphene layers on polar substrates.
I. INTRODUCTION
The high carrier mobility and the large optical absorp-tion in graphene have opened up unique opportunities forthis material in electronics and optoelectronics . Theperformance of graphene in many of these applicationsdepends on the electron-hole generation and recombina-tion rates in graphene. It is therefore important to under-stand the mechanisms that are responsible for electron-hole generation and recombination in graphene and theassociated time scales. Previously, carrier generation andrecombination rates in graphene due to Auger scatter-ing and impact ionization and due to optical phononemission and absorption have been reported . Thestrong interaction between electrons/holes and plasmonsin graphene has been used to explain observed featuresin the angled resolved photoemission (ARPES) data .In this paper, we present electron-hole generation and re-combination rates due to plasmon emission and absorp-tion. Our results show that the recombination times ofcarriers due to plasmon emission are in the tens of fem-toseconds to hundreds of picoseconds range and dependsensitively on the carrier energy, carrier density, temper-ature, and the plasmon dispersion. The available phasespace for plasmon emission is restricted because of en-ergy and momentum conservation requirements and alsobecause of Pauli’s exclusion principle and carriers nearthe Dirac point have much longer lifetimes comparedto carriers at higher energies. Plasmons in a graphenelayer on a polar substrate (Figure (1b)) hybridize withthe surface optical phonons and this hybridization splitsthe plasmon dispersion into two branches . We alsopresent generation and recombination rates of carriersdue to plasmon emission and absorption in graphene lay-ers on polar substrates. The results presented here, in thelight of the previous studies , indicate that plasmonemission is the dominant mechanism for carrier recombi-nation in graphene. Our results are expected to be usefulin interpreting experimental observations in ultrafast op-tical studies and in understanding the operation ofgraphene based optoelectronic devices . II. THEORETICAL MODEL FOR SUSPENDEDGRAPHENE
We first consider a graphene sheet in the plane z = 0sandwiched by media with free-space permittivity (Fig-ure (1a)), as in the case of suspended graphene . Theelectron energy dispersion are given by E s ( ~k ) = s ¯ hvk ,where s equals +1 and -1 for conduction and valencebands, respectively. The dispersion for the plasmons ingiven by ǫ ( q, ω ) = 0, where , ǫ ( q, ω ) = 1 − e ǫ o q Π( q, ω ) (1)The electron-hole propagator Π( q, ω ) is ,Π( q, ω ) = 4 X s,s ′ Z d ~k (2 π ) |h ψ s ′ ~k + ~q ( ~r ) | e i~q.~r | ψ s~k ( ~r ) i| × f s ( ~k ) − f s ′ ( ~k + ~q )¯ hω + E s ( ~k ) − E s ′ ( ~k + ~q ) + iη (2)The matrix element between the Bloch functions in theabove expression equals , |h ψ s ′ ~k + ~q ( ~r ) | e i~q.~r | ψ s~k ( ~r ) i| = 12 " ss ′ k + q cos( θ ) | ~k + ~q | (3)Here, θ is the angle between ~k and ~q . To calculate the re-combination and generation rates, we consider a plasmonwave with the electric field given by, ~E ( ~r, z, t ) = 12 (ˆ q ± i ˆ z ) E o e ∓| ~q | z e i~q.~r − iω ( q ) t + c.c. (4)The transition rate for an electron in the conduction bandto go into the valence band via stimulated emission of aplasmon of wavevector ~q is given by the Fermi’s GoldenRule, 1 τ ~k = 2 π ¯ h |h ψ − ~k − ~q ( ~r ) | e − i~q.~r | ψ + ~k ( ~r ) i| e | E o | q × (1 − f − ( ~k − ~q )) δ ( E + ( ~k ) − E − ( ~k − ~q ) − ¯ hω ( q )) (5)The energy density W of the plasmon wave has contribu-tions from both the field as well as the kinetic energy ofthe carriers. Assuming no plasmon dissipation, the totalenergy density can be found from the complex electro-magnetic energy theorem , W = W F + W KE = ǫ o q | E o | − | E o | ℑ ( ∂σ ( q, ω ) ∂ω (cid:12)(cid:12)(cid:12)(cid:12) ω ( q ) ) (6)Since the conductivity σ ( q, ω ) is related to the dielectricconstant ǫ ( q, ω ) as, ǫ ( q, ω ) = 1 + i qσ ( q, ω )2 ǫ o ω (7)the expression for the energy density W becomes, W = ǫ o q | E o | ℜ ( ω ∂ǫ ( q, ω ) ∂ω (cid:12)(cid:12)(cid:12)(cid:12) ω ( q ) ) (8) W must also equal n ( ~q )¯ hω ( q ) /A , where n ( ~q ) is the num-ber of plasmons in the mode ~q and A is the area of thecrystal. Therefore, using (5) and (8), the lifetime of anelectron in the conduction band due to both stimulatedand spontaneous emission into all plasmon modes be-comes,1 τ ~k = 2 π ¯ h Z d ~q (2 π ) ( n ( ~q ) + 1)(1 − f − ( ~k − ~q )) × e ǫ o q " − k − q cos( θ ) | ~k − ~q | × ¯ hδ ( E + ( ~k ) − E − ( ~k − ~q ) − ¯ hω ( q )) ℜ ( ∂ǫ ( q, ω ) ∂ω (cid:12)(cid:12)(cid:12)(cid:12) ω ( q ) ) (9)The recombination and generation rates, R and G (units: -s) due to plasmon emission and absorption can bewritten as, R = 8 π Z d ~k (2 π ) Z d ~q (2 π ) ( n ( ~q ) + 1) × f + ( ~k )(1 − f − ( ~k − ~q )) × e ǫ o q " − k − q cos( θ ) | ~k − ~q | × δ ( E + ( ~k ) − E − ( ~k − ~q ) − ¯ hω ( q )) ℜ ( ∂ǫ ( q, ω ) ∂ω (cid:12)(cid:12)(cid:12)(cid:12) ω ( q ) ) (10) G = 8 π Z d ~k (2 π ) Z d ~q (2 π ) n ( ~q ) × (1 − f + ( ~k )) f − ( ~k − ~q ) o ! " suboo o Graphene GrapheneSubstrate d Plasmon Plasmon p=np>>n ! " kE ! " kE (a) (b)(c) (d) FIG. 1: (a) A suspended graphene sheet. (b) A graphenesheet on a polar substrate. (c) Electron-hole recombinationvia plasmon emission in p-doped graphene ( p >> n ). (d)Electron-hole recombination via plasmon emission in photoex-cited graphene ( n = p ). × e ǫ o q " − k − q cos( θ ) | ~k − ~q | × δ ( E + ( ~k ) − E − ( ~k − ~q ) − ¯ hω ( q )) ℜ ( ∂ǫ ( q, ω ) ∂ω (cid:12)(cid:12)(cid:12)(cid:12) ω ( q ) ) (11)In thermal equilibrium, the plasmon number n ( ~q ) equalsthe Bose factor (exp(¯ hω ( q ) /KT ) − − . Similar resultscan be obtained starting from the self-energy of an elec-tron in the conduction band. Assuming thermal equilib-rium, and using the imaginary-time Green’s function ap-proach, the relevant contribution to the conduction bandelectron self-energy can be written as , P ( ~k, iω n ) = − Z d ~q (2 π ) " − k − q cos( θ ) | ~k − ~q | × β ¯ h X m e / ǫ o qǫ ( q, iν m ) G ( ~k − ~q, iω n − iν m ) (12)where G ( ~k − ~q, ω n − iν m ) is the valence band Green’s func-tion. The summation over the Matsubara frequencies canbe performed by first isolating the pole coming from thezero of ǫ ( q, iν m ) in the denominator at the plasmon fre-quency. Finally, if one excludes from (12) contributionscoming from plasmon absorption processes and then cal-culates the lifetime of the conduction electron using theexpression ,1 τ ~k = − h ℑ nP ( ~k, ( E + ( ~k ) − E f ) / ¯ h + iη ) o (13)then the result obtained is identical to the one given ear-lier in (9). It should be mentioned here that focusingon the collective excitation pole coming from the zero of E n e r g y ( e V ) p=10 cm −2 p=10 cm −2 T=30K
FIG. 2: The plasmon dispersion (solid) in a p-doped grapheneis shown for different hole densities. The dashed curve repre-sents ¯ hvq . T = 30 K . ǫ ( q, iν m ) allows one to calculate the interband scatteringrate due to electron-plasmon interaction. Contributionsfrom other processes, such as Auger scattering and im-pact ionization , are therefore excluded. From the re-sults obtained above it follows that the electron-plasmoninteraction in graphene can be approximately describedby the following Hamiltonian in the second quantizedform,ˆ H el − pl = X s,s ′ ,σ,~k,~q M s,s ′ ,~k,~q (cid:16) ˆ b ~q + ˆ b †− ~q (cid:17) ˆ c † s ′ ,σ,~k + ~q ˆ c s,σ,~k (14)where, ˆ b ~q and ˆ c s,σ,~k are the plasmon and the electrondestruction operators, respectively, σ stands for differentspins and valleys, and the coupling constant M s,s ′ ,~k,~q isgiven by, (cid:12)(cid:12)(cid:12) M s,s ′ ,~k,~q (cid:12)(cid:12)(cid:12) = e ¯ h ǫ o q A " ss ′ k + q cos( θ ) | ~k + ~q | ℜ ( ∂ǫ ( q, ω ) ∂ω (cid:12)(cid:12)(cid:12)(cid:12) ω ( q ) ) (15)Here, θ is the angle between ~k and ~q and A is the area ofthe graphene crystal. A. Results and Discussion
The plasmon dispersion is first found numerically us-ing the expression for Π( q, ω ) in (2). The recombina-tion and generation rates and lifetimes are then calcu-lated using (9), (10), and (11). The dominant contribu-tion to the propagator in (2) comes from the intrabandpart. The interband part modifies the plasmon disper-sion slightly and also imparts an imaginary part to the −3 −2 −1 Electron Energy (meV) T i m e ( p s ) Solid: T = 30KDashed: T = 300Kp=10 cm −2 p=10 cm −2 FIG. 3: The calculated spontaneous emission lifetime of anelectron in the conduction band is plotted as a function of theelectron energy for different hole densities and temperaturesin p-doped graphene. plasmon frequency. If the interband contribution is ig-nored the error in the calculated plasmon frequency hasbeen found to be generally small (less than 10%) for smallplasmon wavevectors. This small error comes with theenormous simplicity of having to find zeroes of ǫ ( q, ω ) ononly the real frequency axis and therefore this approachhas been adopted in numerical simulations. The resultswe present are also not self-consistent in the sense thatthe quasiparticle density of states have been assumedto be that of the non-interacting electron system. It isknown that electron-plasmon interaction can modify thequasiparticle density of states and generate plasmaronbands . However, the modification of the quasiparticledensity of states is expected to be small for the elec-tron and hole densities considered in this paper. Theaverage recombination and generation times are definedas, τ − R = R/ min( n, p ), and τ − G = G/ min( n, p ), respec-tively, where n and p are the electron and hole densities.Figure (2) shows the plasmon dispersion in p-dopedgraphene for different hole densities. Figure (3) showsthe calculated lifetime of an electron in the conductionband due to spontaneous plasmon emission as a functionof the electron energy for different hole densities and tem-peratures in p-doped graphene (Figure (1c)). Note thatthe electron-hole symmetry in graphene implies that thehole lifetimes in n-doped graphene would be identical.Figure (3) shows that conduction electrons with energiesnear the Dirac point have much longer lifetimes com-pared to the electrons at higher energies. This trend canbe understood as follows. Energy and momentum con-servation require that an electron with wavevector ~k canemit a plasmon with wavevector ~q only if,12 (cid:18) ω ( q ) v − q (cid:19) ≤ k ≤ (cid:18) ω ( q ) v + q (cid:19) (16)The shaded region in Figure (4) shows the allowed val-ues of the wavevector of the emitted plasmon as a func-tion of the electron energy assuming p = 10 cm − and T = 30 K . As the electron energy becomes smallerthan ∼
60 meV, the allowed phase space for the plas-mon wavevectors shrinks significantly for small wavevec-tors. The probabilities of emission of plasmons of largewavevectors are small because the energies of such plas-mons are large and the resulting final electron states deepinside the valence band are already occupied by valenceelectrons. In addition, numerical simulations show that ℜ n ∂ǫ ( q, ω ) /∂ω | ω ( q ) o in (9) becomes large when ω ( q ) ap-proaches qv , which happens for very large wavevectorsand this also reduces the probability of emission of largewavevector plasmons. Figure (3) shows that the electronspontaneous emission lifetimes can range from values assmall as 10 fs to hundreds of picoseconds. Figure (5)shows the average minority carrier (electron) recombina-tion time τ R plotted as a function of the minority carrierdensity for different temperatures in p-doped graphene( p = 10 cm − ). As expected from the results in Figure(3), the average recombination time decreases with theincrease in the temperature because the minority carrierdistribution spreads to higher energies.In many optical studies and in graphene based op-toelectronic devices , photoexcitation followed byrapid thermalization results in a an equal number of ther-mally distributed electrons and holes in an otherwise nearintrinsic graphene layer (Figure (1d)). It is therefore im-portant to understand the recombination and generationtimes in such situations. Figure (6) shows the recom-bination times τ R (solid) and the generation times τ G (dashed) plotted as a function of the electron and holedensity (assumed to be equal) for different temperatures.The number of plasmons n ( ~q ) in different modes is as- FIG. 4: The shaded region shows the allowed values of thespontaneously emitted plasmon wavevectors as a functionof the electron energy in the conduction band for p-dopedgraphene. p = 10 cm − and T = 30 K −3 −2 −1 Minority Carrier Density (cm −2 ) T i m e ( p s ) p = 10 cm −2 T = 30K, 100K, 300K, 900K
FIG. 5: The average minority carrier (electron) recombinationtime τ R is plotted as a function of the minority carrier den-sity for different temperatures in p-doped graphene ( p = 10 cm − ). The arrow indicates curves for increasing values ofthe temperature ( T = 30 , , , −3 −2 −1 Carrier Density (cm −2 ) T i m e ( p s ) Solid: Recombination TimeDashed: Generation TimeT = 30, 100, 300, 900 Kn=p
FIG. 6: The recombination times τ R (solid) and the genera-tion times τ G (dashed) are plotted as a function of the electronand hole density (assumed to be equal) for different tempera-tures. The arrows indicate curves for increasing values of thetemperature ( T = 30 , , , sumed to be given by the Bose factor. This assumptionmay not be valid in a non-equilibrium situation imme-diately following photoexcitation. Figure (7) shows therecombination times τ R (solid) and the generation times τ G (dashed) plotted as a function of the temperature fordifferent electron and hole densities. Figures (6) and (7)show that the recombination times can be much smallerthan a picosecond for carrier densities larger than 10 cm − at all temperatures. Figures (6) and (7) show thatthe generation times can also be very short and this im-plies that carrier generation cannot be ignored in experi-ments where a hot carrier distribution is created via pho- −2 −1 Temperature (K) T i m e ( p s ) Solid: Recombination TimeDashed: Generation Timen=p=10 ,10 ,10 ,10 cm −2 FIG. 7: The recombination times τ R (solid) and the genera-tion times τ G (dashed) are plotted as a function of the tem-perature for different electron and hole densities (assumed tobe equal). The arrows indicate curves for increasing values ofthe carrier density ( n = p = 10 , , , cm − ). toexcitation in ultrafast optical studies . III. THEORETICAL MODEL FOR GRAPHENEON POLAR SUBSTRATES
The results presented above suggest that it mightbe possible to alter the plasmon-assisted recombina-tion and generation rates in graphene by altering thedielectric environment thereby modifying the strengthof the Coulomb interaction . Specifically, a substratewith a large dielectric constant could potentially reducethe recombination and generation rates. However, po-lar materials with large dielectric constants have sur-face optical phonon modes that couple strongly with thegraphene plasmons . To study this further, we considera graphene sheet at a distance d away from a polar sub-strate (Figure (1b)). The dielectric constant of the sub-strate is assumed to be given by the expression, ǫ sub ( ω ) = ǫ sub ( ∞ ) (cid:18) ω − ω LO ω − ω T O (cid:19) (17)Here, ǫ sub (0) /ǫ sub ( ∞ ) = ω LO /ω T O . The surface opti-cal phonon frequency ω SO is obtained by setting ǫ sub ( ω )equal to -1, and equals, ω SO = ω T O s ǫ sub (0) + 1 ǫ sub ( ∞ ) + 1 (18)The dielectric constant ǫ ( q, ω ) of the graphene sheet canbe found by placing a test charge in the sheet and findingthe resulting potential. The result is, ǫ ( q, ω ) = 12 + 12 (cid:20) ( ǫ sub ( ω ) + 1) e qd + ( ǫ sub ( ω ) − ǫ sub ( ω ) + 1) e qd − ( ǫ sub ( ω ) − (cid:21) E n e r g y ( e V ) T=30K p=10 cm −2 h ω SO p=10 cm −2 FIG. 8: The dispersion of the coupled plasmon-phonon mode(solid) in a p-doped graphene sheet on a SiC substrate isshown for different hole densities. The dispersion splits intotwo branches. The dashed curve represents ¯ hvq . T = 30 K . − e ǫ o q Π( q, ω ) (19)The dispersion of the coupled plasmon-phonon longitudi-nal mode can be found as before by setting ǫ ( q, ω ) equalto zero. Now one finds two longitudinal collective modes.In the q → ω ( q ) → q →
0, and the higher frequencymode is phonon-like with ω ( q ) → ω SO as q →
0. For largewavevectors, the lower frequency mode disappears intothe electron-hole continuum while the higher frequencymode becomes plasmon-like with ω ( q ) → qv as q → ∞ .As an example, we consider the technologically relevantcase of a graphene layer on a Silicon Carbide (SiC) sub-strate . The values of different parameters are asfollows: d = 5 Angstroms, ¯ hω LO = 120 meV, ¯ hω T O = 98meV, and ǫ ∞ = 6 . . These give ¯ hω SO ≈
117 meV.Figure (8) shows the dispersions of the coupled plasmon-phonon modes for a p-doped graphene sheet on a SiCsubstrate for different hole densities. Comparing Figures(2) and (8), it can be seen that plasmon-phonon cou-pling significantly modifies the dispersion and this hasrecently been verified experimentally . The recombi-nation and generation rates can be obtained using thesame expressions as those given in (9), (10), and (11)with the exception that contributions from both branchesof the dispersion must be included. Therefore, surfaceoptical phonons of the polar substrate provide an addi-tional channel for carrier recombination and generation.It should be mentioned here that large wavevector surfaceoptical phonon modes can also cause intervalley recombi-nation and generation processes . However, the squareof the coupling matrix element between the surface opti-cal phonons and the carriers is proportional to , e ǫ o q e − qd ¯ hω SO (cid:18) ǫ sub ( ∞ ) + 1 − ǫ sub (0) + 1 (cid:19) (20) −3 −2 −1 Electron Energy (meV) T i m e ( p s ) Solid: T = 30KDashed: T = 300K p=10 cm −2 p=10 cm −2 FIG. 9: The calculated spontaneous emission lifetime of anelectron in the conduction band is plotted as a function of theelectron energy for different hole densities and temperaturesin p-doped graphene on a SiC substrate. −2 −1 Minoriy Carrier Density (1/cm −2 ) T i m e ( p s ) T = 30K, 100K, 300K, 900Kp = 10 cm −2 FIG. 10: The average minority carrier (electron) recombina-tion time τ R is plotted as a function of the minority carrierdensity for different temperatures in p-doped graphene ( p =10 cm − ) on a SiC substrate. The arrow indicates curves forincreasing values of the temperature ( T = 30 , , , and becomes small for the large wavevectors needed forthe intervalley transitions in graphene.Therefore, inter-valley processes will be ignored here. A. Results and Discussion
Figure (9) shows the calculated lifetime of an electronin the conduction band due to spontaneous emission as afunction of the electron energy for different hole densitiesand temperatures for a p-doped graphene sheet on a SiCsubstrate. Figure (9) displays the same general trends as −2 −1 Carrier Density (cm −2 ) T i m e ( p s ) Solid: Recombination TimesDashed: Generation TimesT=30, 100, 300, 900Kn=p
FIG. 11: The recombination times τ R (solid) and the gen-eration times τ G (dashed) are plotted as a function of theelectron and hole density (assumed to be equal) for differenttemperatures for a graphene sheet on a SiC substrate. The ar-rows indicate curves for increasing values of the temperature( T = 30 , , , does Figure (3) in the case of a suspended graphene sheet.However, lifetimes are shorter for the low energy electronsin the case of graphene on SiC. For electrons with ener-gies near the Dirac point, recombination is entirely dueto the lower frequency branch of the dispersion whichfacilitates interband transitions better than the plasmondispersion in suspended graphene. The sharp peaks seenin Figure (9) occur when the lifetimes due to the lowerfrequency branch of the dispersion are becoming longerwith the electron energy while lifetimes due to the up-per frequency branch are becoming shorter. As in thesuspended graphene case, the spontaneous emission life-times can range from tens of femtoseconds to hundredsof picoseconds. Figure (10) shows the minority carrier(electron) recombination time τ R plotted as a function ofthe minority carrier density for different temperatures ina p-doped graphene ( p = 10 cm − ) on a SiC substrate.Compared to the suspended graphene case (Figure 5), therecombination times for graphene on SiC are shorter forsmall minority carrier densities. Next, we consider thecase when the electron and hole densities are the same(as is the situation in photoexcitation experiments). Fig-ure (11) shows the recombination times τ R (solid) andthe generation times τ G (dashed) plotted as a functionof the electron and hole density (assumed to be equal)for different temperatures for a graphene sheet on a SiCsubstrate. The recombination and generation times ingraphene on SiC are generally of the same order as inthe case of suspended graphene discussed earlier. Therole of the higher dielectric constant of the SiC substratein reducing plasmon-assisted recombination and genera-tion rates, compared to suspended graphene, is compen-sated by the presence of surface optical phonons whichnot only modify the plasmon dispersion but also providean additional channel for recombination and generation. IV. CONCLUSION
In this paper we have presented electron-hole re-combination and generation times due to spontaneousand stimulated emission and absorption of plasmons ingraphene. Our results indicate that plasmon assisted re-combination times in graphene can vary over a wide rangeof values ranging from tens of femtoseconds to hundredsof picoseconds. In many proposed and demonstrated op-toelectronic devices , the plasmon-assistedrecombination and generation rates could be fast enoughto significantly impact device performance.
Acknowledgments
The authors acknowledge helpful discussions with PaulL. McEuen, Michael G. Spencer, and Jiwoong Park, andacknowledge support from the National Science Founda-tion (monitor Eric Johnson), the DARPA Young FacultyAward, the MURI program of the Air Force Office of Sci-entific Research (monitor Harold Weinstock), the Officeof Naval Research (monitor Paul Makki), and the Cor-nell Material Science and Engineering Center (CCMR)program of the National Science Foundation. ∗ Electronic address: [email protected] K. S. Novoselov et. al., Nature, , 197 (2005). K. S. Novoselov et. al., Science, , 666 (2004). R. Saito, G. Dresselhaus, M. S. Dresselhaus,
Physical Prop-erties of Carbon Nanotubes , Imperial College Press, Lon-don, UK (1999). W. De Heer et. al., Science, , 1191 (2006). I. Meric, M. Y. Han, A. F. Young, B. Ozyilmaz, P. Kim,K. L. Shepard, Nature Nanotechnology, , 654 (2008). J. R. Williams, L. DiCarlo, C. M. Marcus, Science, ,638 (2007). F. Rana, IEEE Trans. Nanotech., , 91 (2008). V. Ryzhii, M. Ryzhii, T. Otsuji, J. Appl. Phys., ,083114 (2007). F. Xia, T. Mueller, Y. Lin, A. V. Garcia, P. Avouris, Na-ture Nanotechnology, , 839 (2009). F. T. Vasko, V. Ryzhii, Phys. Rev. B, , 195433 (2008). V. Ryzhii, M. Ryzhii, V. Mitin, T. Otsuji, J. Appl. Phy., , 054512 (2010). F. Bonaccorso, Z. Sun, T. Hasan, A. C. Ferrari, Nature, ,611 (2010). T. Mueller, F. Xia, P. Avouris, Nature Photonics, , 297(2010). M. Ryzhii, V. Ryzhii, T. Otsuji, V. Mitin, M. S. Shur,Phys. Rev. B, , 075419 (2010). V. Ryzhii, A. A. Dubinov, T. Otsuji, V. Mitin, M. S. Shur,J. Appl. Phys., , 054505 (2010). V. Ryzhii, M. Ryzhii, T. Otsuji, J. Appl. Phys., ,083114 (2007). A. Bostwick, F. Speck, T. Seyller, K. Horn, M. Polini, R.Asgari, A. H. MacDonald, E. Rotenberg, Science, , 999(2010). A. Bostwick, T. Ohta, T. Seyller, K. Horn, E. Rotenberg,Nature Physics, , 36 (2007). M. Polini, R. Asgari, G. Borghi, Y. Barlas, T. Pereg-Barnea, A. H. MacDonald, Phys. Rev. B, , 081411(R)(2008). E. H. Hwang, S. Das Sarma, Phys. Rev. B, , 081412(R)(2008). F. Rana, Phys. Rev. B, , 155431 (2007). F. Rana, P. A. George, J. H. Strait, J. Dawlaty, S. Shiv-araman, M. Chandrashekhar, M. G. Spencer, Phys. Rev.B, , 115447 (2009). R. J. Koch, T. Seyller, J. A. Schaefer, B, , 201413(R)(2010). E. H. Hwang, R. Sensarma, S. Das Sarma, Phys. Rev. B, , 195406 (2010). P. A. George, J. Strait, J. Dawlaty, S. Shivaraman, MvsChandrashekhar, F. Rana, M. G. Spencer, Nano Lett., ,4248 (2008). J. M. Dawlaty, S. Shivaraman, Mvs Chandrashekhar, F.Rana, M. G. Spencer, Appl. Phys. Lett., , 042116(2008). Chun Hung Lui, Kin Fai Mak, Jie Shan, Tony F. Heinz,Phys. Rev. Lett., , 127404 (2010). Kin Fai Mak, Chun Hung Lui, Tony F. Heinz, Appl. Phys.Lett., , 221904 (2010). D. Sun, Z. K. Wu, C. Divin, X. Li, C. Berger, W. A. deHeer, P. N. First, T. B. Norris, Phys. Rev. Lett., ,157402 (2008). R. W. Newson, J. Dean, B. Schmidt, and H. M. van Driel,Opt. Express, , 2326 (2009); H. N. Wang, J. H. Strait, P. A. George, S. Shivaraman, V.B. Shields, M. Chandrashekhar, J. Hwang, F. Rana, M. G.Spencer, C. S. Ruiz-Vargas, and J. Park, Appl. Phys. Lett.96, 081917 (2010). H. Choi, F. Borondics, D. A. Siegel, S. Y. Zhou, M. C.Martin, A. Lanzara, R. A. Kaindl, Appl. Phys. Lett., ,172102 (2009). K.I. Bolotin, K.J. Sikes, Z. Jianga, M. Klima, G. Fuden-berg, J. Hone, P. Kim, H.L. Stormer, Solid State Com-mun., , 351 (2008). E. H. Hwang, S. Das Sarma, Phys. Rev. B, , 205418(2007). X. F. Wang, T. Chakraborty, Phys. Rev. B, , 033408(2007). D. H. Staelin, A. W. Morgenthaler, J. A. Kong,
Electro-magnetic Waves ,Prentice Hall, NJ (1998). G. D. Mahan,
Many Particle Physics , Plenum Press, NY(1990). X. Xu, N. M. Gabor, J. S. Alden, A. M. Van Der Zande,P. L. McEuen, Nano Lett., , 562 (2010). J. Park, Y. H. Ahn, C. S. Ruiz-Vargas, Nano Lett., , 1742(2009). H. Nienhaus, T. U. Kampen, W. Monch, Surf. Sci., 324,L528 (1995). D. Jena, A. Konar, Phys. Rev. Lett., , 136805 (2007). S. Q. Wang and G. D. Mahan, Phys. Rev. B,6