Ultrafast Coulomb-Induced Intervalley Coupling in Atomically Thin

Apr 18, 2016 - Long-Lived Valley Polarization of Intravalley Trions in Monolayer WSe 2. Akshay Singh , Kha Tran , Mirco Kolarczik , Joe Seifert , Yipi...
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Ultrafast Coulomb-Induced Intervalley Coupling in Atomically Thin WS2 Robert Schmidt,† Gunnar Berghaü ser,‡ Robert Schneider,† Malte Selig,§ Philipp Tonndorf,† Ermin Malić,‡ Andreas Knorr,§ Steffen Michaelis de Vasconcellos,† and Rudolf Bratschitsch*,† †

Institute of Physics and Center for Nanotechnology, University of Münster, 48149 Münster, Germany Department for Applied Physics, Chalmers University of Technology, SE-41296, Gothenburg, Sweden § Department for Theoretical Physics, Technical University Berlin, 10623 Berlin, Germany ‡

S Supporting Information *

ABSTRACT: Monolayers of semiconducting transition metal dichalcogenides hold the promise for a new paradigm in electronics by exploiting the valley degree of freedom in addition to charge and spin. For MoS2, WS2, and WSe2, valley polarization can be conveniently initialized and read out by circularly polarized light. However, the underlying microscopic processes governing valley polarization in these atomically thin equivalents of graphene are still not fully understood. Here, we present a joint experiment−theory study on the ultrafast timeresolved intervalley dynamics in monolayer WS2. Based on a microscopic theory, we reveal the many-particle mechanisms behind the observed spectral features. We show that Coulombinduced intervalley coupling explains the immediate and prominent pump−probe signal in the unpumped valley and the seemingly low valley polarization degrees typically observed in pump−probe measurements compared to photoluminescence studies. The gained insights are also applicable to other light-emitting monolayer transition metal dichalcogenides, such as MoS2 and WSe2, where the Coulomb-induced intervalley coupling also determines the initial carrier dynamics. KEYWORDS: 2D materials, transition metal dichalcogenides, ultrafast valley dynamics, screened Coulomb matrix elements

A

prevents easy access to the underlying physical processes with picosecond photoluminescence spectroscopy. Recent femtosecond pump−probe experiments on monolayer MoS2 and WS2 indicated that renormalization effects have to be considered for the interpretation of the pump−probe spectra.22,23 Valley-resolved measurements have shown a multitude of spectral features with different interpretations.24−27 Most surprisingly, a strong signal in the unpumped K′ valley immediately after pumping the K valley has been found. To explain this strong signal, Mai et al.24,25 assumed that an electron population created by the pump pulse in one valley spreads over both valleys due to the degeneracy of the conduction bands forming “dark exciton” states. These states would give rise to a positive signal in the unpumped valley due to Pauli blocking. Another interpretation involved the creation of biexcitons with binding energies on the order of 40 meV.26,27 In this case, fast intervalley scattering processes of excitons due to the Coulomb exchange interaction would create a strong positive signal in the unpumped valley due to absorption bleaching. In addition, excitons created by the pump pulse in

tomically thin transition metal dichalcogenides (TMDCs) such as MoS2, MoSe2, WS2, and WSe2 have recently received much attention as a promising material system for optoelectronic devices. They share favorable physical properties with graphene,1 such as high flexibility, mechanical strength, transparency, and atomic thickness. In addition, semiconducting TMDCs possess a direct band gap in the visible to nearinfrared regime.2,3 This property renders them promising for ultrathin p−n junctions, solar cells, or light-emitting diodes.4−7 Interestingly, the direct band gap occurs both at the K and K′ points of the Brillouin zone.2 It has been demonstrated that together with the absence of inversion symmetry this material property leads to the emission of polarized photoluminescence after excitation with circularly polarized light.8−13 This longsought initialization and readout of valley polarization opens up exciting possibilities for electronics and computing based on the valley degree of freedom, originally proposed for graphene.14−16 To reveal the physical mechanisms behind the creation and destruction of valley polarization, time-resolved experiments and microscopic calculations are needed. Recent experiments have shown that the valley polarization decays on the order of a few picoseconds,17−19 which is attributed to strong Coulomb exchange interactions in these materials.19−21 This fast decay © 2016 American Chemical Society

Received: November 19, 2015 Revised: February 25, 2016 Published: April 18, 2016 2945

DOI: 10.1021/acs.nanolett.5b04733 Nano Lett. 2016, 16, 2945−2950

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Figure 1. Ultrafast valley-resolved dynamics of the A exciton in monolayer WS2. (a) Measured time-, polarization-, and energy-resolved differential transmission of the A exciton for probing and pumping the same (SCP) or different (OCP) valley via circularly polarized laser pulses. (b) Differential transmission spectra for four different time delays for same and opposite circular polarization of the probe pulses.

valley) polarization (see Supporting Information for details). We refer to the measurement pumping and probing the same K valley as same circular polarization (SCP), while the measurement probing the K′ valley is named opposite circular polarization (OCP). The measured differential transmission signal (DTS) is given by

one valley would increase the probability for the probe pulse to create intervalley biexcitons in the unpumped valley. This effect would result in an additional red-shifted negative differential transmission signal, giving rise to an overall dispersive shape. Here, we measure differential transmission spectra obtained by valley and spectrally resolved femtosecond pump−probe spectroscopy and directly compare the experimental data to microscopic calculations based on semiconductor Bloch equations. We show that the prominent spectral features in both valleys directly after the pump pulse can be explained by Coulomb-induced band gap renormalization. Furthermore, we find a moderate, impurity-induced intervalley carrier transfer during the optical excitation. It decays with the coherent quantities on a femtosecond time scale and becomes more prominent in the strong excitation regime, where the direct excitation of carriers is strongly suppressed via Pauli blocking. A single layer of WS2 is mechanically exfoliated from a bulk single crystal onto a borosilicate glass substrate. The monolayer is identified by photoluminescence, transmission, and atomic force microscopy measurements. In the linear absorption spectrum (Supplementary Figure S1b) we find strong resonances of the A (2.03 eV) and B excitons (2.42 eV), while no trion signal is visible. The atomically thin semiconductor is optically excited by a left-handed circularly polarized pump pulse with typical fluences of 6 μJ/cm2. The optical excitation is spectrally tuned to the high-energy side of the A exciton (2.10 eV) to selectively create excitons in the K valley with a density of 2 × 1011 cm−2. To probe the exciton dynamics in both valleys we use a broadband probe pulse (1.9− 2.13 eV) with either left (K valley) or right-handed circular (K′

T (ω , Δt ) − TOff (ω , Δt ) ΔT (ω , Δt ) = On T TOff (ω , Δt ) ≈

αOff (ω , Δt ) − αOn(ω , Δt ) 1 − αOff (ω , Δt )

(1)

where TOn(αOn) and TOff(αOff) is the transmission (absorption) with and without optical excitation and ω and Δt denote the probing energy and the time delay between pump and probe pulse. For monolayer WS2 the reflectivity is 1 order of magnitude smaller than the absorption at energies close to the A exciton resonance.28 Therefore, a measurement of transmission directly provides information on the absorption of the monolayer (eq 1). Figure 1a shows the time evolution of the differential transmission spectra (DTS) for the same (SCP) and opposite circular polarization (OCP) of pump and probe pulses. The SCP measurement reveals a positive Gaussian-shaped signal at zero time delay, while it transforms into a slightly dispersive profile at delays longer than 1 ps (red curves in Figure 1b). For delay times below 10 ps, the OCP signal strongly differs from the SCP spectra. The OCP spectra always exhibit a clear dispersive profile with a pronounced negative component at energies smaller than the A exciton resonance (blue curves in 2946

DOI: 10.1021/acs.nanolett.5b04733 Nano Lett. 2016, 16, 2945−2950

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Nano Letters Figure 1b). At longer time delays we observe two effects. There is an overall decrease of the pump−probe signal for both valleys with a time constant of 40 ps due to the decay of the A excitons. The difference between the SCP and OCP signal vanishes within about 10 ps due to intervalley scattering, which indicates that the pump-induced valley polarization is lost after this time.25 As a result, the degree of valley polarization in a time-integrated experiment of this sample under the same conditions would be below 100%. In the following discussion, we concentrate on the spectral features at zero time delay. To explain the measured signals, we perform microscopic calculations based on the density matrix formalism. In general, the optical properties of atomically thin transition metal dichalcogenides are dominated by pronounced excitonic transitions stemming from the K and K′ valleys.17,29−34 Thus, we restrict our theoretical investigation to these characteristic points of the TMDC Brillouin zone. According to the performed experiments, we focus on the optical response of monolayer WS2 after resonantly pumping the energetically lowest A exciton in one of the two valleys. Electron−hole pairs are generated in the K or K′ valley using left- or right-handed circularly polarized light, respectively. The corresponding A excitons in both valleys are energetically degenerate but differ in spin and valley number. To calculate the transient optical response on a microscopic footing, we solve the Heisenberg’s equation of motion iℏO = [H,O]− for single-particle quantities ρi,j = ⟨a†i aj⟩ with a†i creating and aj annihilating a particle in the state i and j, respectively. The many-particle Hamilton operator consists of the freecarrier contribution H0, the light−matter interaction Hlm, the carrier−disorder coupling HD, and the Coulomb interaction HC. Here, we focus on the low-excitation regime, where we can truncate the many-particle hierarchy problem on the Hartree− Fock level.35,36 This yields TMDC Bloch equations37,38 for the s † microscopic polarization pcvξ kA,kB = ⟨ackA,ξs avkB,ξs⟩ and the charge λξs carrier density σkA,kB = ⟨a†λkA,ξsaλkB,ξs⟩. Then, we perform an s excitonic expansion of these equations using the relation pnξ Q = cvξs nξs nξs ∑qpq,Qϕq with ϕq as the excitonic wave function of the state n.39,40 Following this scheme, we find the coupled TMDC Bloch equations in the excitonic expansion:

The quantities μ λ ̅ λ =

∑ DQnl′ξ pQlξ′− Q s

s

lQ ′

+

∑

nmlξs mξs (Γ Q e σe

nmlξs mξs lξs + ΓQ h σh )pQ

m,l

+

nmlξ ξ ̅ mξ ̅ lξ nmlξ ξ ̅ mξ ̅ ∑ (Y Qe σe + Y Q h σh )pQ, s s s

s s

s

s

s

m,l

−

∑ (X Qnlξ ξ ̅ δm,l − X Qnmle ξ ξ ̅ σemξ s s̅

s s̅

s

nmlξsξ s̅ ̅ mξs lξ s̅ ̅ − XQ σh )pQ h

m,l

−

∑

nξsξ s̅ ̅ ZQ ′ Q,e,h, p

(2)

Q ′≠ Q nξs σ̇e(h) =

2 Im[p0nξs (Ωnξs) * + ℏ

∑ XQnξξ′ ̅ (pQnξ′ ̅ )*pQnξ′ ] s̅

Q′

s̅

are determined by the effective

s and holes msh with spin s, while σnξ masses of electrons h/e ≡ nξs σh/eQ=0 correspond to electron (e) and hole (h) occupations projected into the excitonic basis in analogy to the microscopic polarization. The quantities including a nonzero center-of-mass s momentum σnξ h/eQ≠0 corresponding to intraband polarizations sξ̅s ̅ are summarized in the term Znξ Q′Q,e,h,p. They are expected to contribute to a dephasing of the microscopic polarization. The dephasing rate is accounted for by γhom Q which also includes the radiative dephasing and the higher-order carrier-phonon and carrier-carrier scattering terms.41 The dephasing of the microscopic polarization leads to an homogeneous broadening of the absorption. For monolayer WSe2, the homogeneous broadening was recently found to be in the range of 3 meV at a temperature of 4 K, while the total line width was at least 1 order of magnitude larger due to inhomogeneous broadening.41 Furthermore, a temperature dependent broadening of the excitonic resonances in monolayer WSe2 has been observed, which at room temperature is in the range of tens of meV.42 In addition, we have calculated that exciton−phonon coupling leads to an increase of the homogeneous line width γhom up to 20 meV at room temperature.43 Therefore, we choose γhom = 20 meV for our calculations. The absorption line width measured at a temperature of 4 K is on the order of 20 meV for our sample (data not shown), which as mentioned above is ascribed to inhomogeneous broadening γinhom, following the argumentation of Moody et al.41 At room temperature, γhom and γinhom add up to a Voigt profile with a line width of 33 meV, which is in good agreement with our measured room temperature absorption spectrum (Figure S1b). In our model, the inhomogeneous broadening s has been considered via the disorder potential Dnlξ Q′ in eq 2. (See Supporting Information for details.) The disorder coupling induces a center of mass motion of the exciton which is driving the intervalley interaction via X-coupling, as discussed below. The Coulomb induced terms are separated in three contributions: (i) the intrinsic intravalley correlation Γ, (ii) the intrinsic intervalley scattering Y which couples states at the K and K′ valley with the same spin, and (iii) the impurityassisted electron−hole exchange coupling X, coupling energetic resonant states in the K and K′ valley. The Coulomb matrix elements are expressed by the Keldysh potential V′Q, which is known to well describe the excitonic properties of atomically thin 2D materials on a specific substrate.29,44−47 The origin of the Y coupling is an overlap of intraband orbital functions, whereas the Coulomb exchange X coupling stems from an overlap of interband orbital functions. It couples resonant states at different valleys (A−A and B−B) and induces a microscopic polarization in the unpumped valley resulting in a carrier transfer. A more detailed discussion of the contribution of the different Coulomb terms and the corresponding matrix elements can be found in the Supporting Information. The numerical solution of the coupled TMDC Bloch equations (eqs 2 and 3) provides access to the time- and energy-dependent microscopic polarization, allowing us to calculate the frequency-dependent optical absorption. Exploiting this knowledge and applying eq 1, we can model the experimentally measured differential transmission spectra for pumping and probing the same (SCP) and different valleys (OCP). We focus on the dynamics directly after optical excitation by the pump pulse. Scattering processes are not included, since they are assumed to give rise to a population

mse

nξs nξs iℏpQ̇nξs = (EQ + Enξs − iγQhom)pQnξs + δQ,0(Ωnξs − Ωeh σe nξs nξs − Ωeh σh ) +

mλ mλ + m λ̅

s

(3)

We have introduced the valley index ξs, with the subindex s for the spin, and the center-of-mass momentum Q that has been separated from the relative momentum q = μλλ̅kA + μλ̅λkB. The excitonic state is given by quantum numbers n, m, and l. 2947

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Nano Letters decay on a time scale of tens of picoseconds.25 To generate a nonequilibrium distribution of excitons, we apply a pump pulse at an energy of 2 eV. The absorbed pump fluence is varied from 0.15 to 4 μJ cm−2 inducing carrier densities of 0.47 × 1012 cm−2 to 12.4 × 1012 cm−2. By comparing the photoluminescence intensities of exciton and trion at low temperature (data not shown), we can estimate the doping level48 of the WS2 monolayer to be on the order of 1011 cm−2. Since the doping level is a free parameter in the microscopic calculations, we have varied this parameter to achieve the best agreement between the experimental and theoretical data. A value of 1 × 1011 cm−2 provides a good agreement between experiment and theory and is used in the calculations. Doping leads to screening of the Coulomb potential, which is included in our model via Lindhard screening. In addition, the carriers induce phase space filling (cf. eq 2), which accounts for Pauli blocking and also renormalizes the band gap energy. Figure 3 presents a direct comparison between the measured and the calculated differential transmission spectra for same (SCP) and opposite circular polarization (OCP) of the pump and the probe pulse for a series of different fluences at zero time delay. Both in experiment and theory, the SCP and OCP amplitudes are in the same order of magnitude. To focus on the spectral shifts, all data are shown normalized in Figure 3. The experimental data is labeled with the pump fluences exciting the monolayer, while for the calculations the exciton density is given, because the relation between the two is nonlinear in the excitation regime used. Both experiment and theory exhibit the same qualitative features: (i) a pronounced OCP signal indicating an efficient intervalley coupling, (ii) a dispersive DTS shape both in the OCP and the SCP case, however more pronounced for OCP, and (iii) a spectral shift of both, the SCP and OCP signal to lower energies with increasing pump fluence. These features stem from the following elementary processes that are schematically illustrated in Figure 2. For pumping and probing the same valley (SCP), optically excited carriers give rise to a change of the band gap energy and lead to a renormalization of the Rabi frequency, causing a net shift of the exciton resonance (Figure 2a, left). Furthermore, the phase space filling factor in eq 2 leading to absorption bleaching significantly increases due to the injection of charge carriers. As a consequence, the shifted and bleached absorption in the excited case results in a slightly dispersive shape consisting of a positive DTS signal at the resonance of the undisturbed system and a small negative component at lower energies (Figure 2b, left). This effect is clearly observed both in the experimental and theoretical SCP data shown in Figure 3. When probing the unpumped K′ valley (OCP), a strong dispersive DTS signal is found. The free-particle picture based solely on the electronic band structure of WS2 fails to explain this feature, where a zero DTS is expected. The appearance of a strong immediate signal in the OCP can be traced back to the Coulomb coupling of both valleys. The strength of this intervalley coupling is given by the Y and X terms of eq 2. In the experiment the carriers are pumped 100 meV above the A exciton. This leads to an off-resonant excitation of A and B excitons, which is enhanced by the broadening of excitonic resonances. The off-resonantly induced population is sufficient for the Y mechanism to couple excitons of the same spins in different valleys, i.e. A and B excitons in the K and K′ valley (Figure 2). In this way, also an energy renormalization of the unpumped valley takes place, giving rise to the pronounced

Figure 2. Coulomb-induced intervalley coupling: (a) Schematic drawing of the optically dominant excitonic A and B transitions in the K and K′ valley. The vertical gray arrows indicate the exciton spins. Coulomb-induced intervalley coupling in TMDC materials gives rise to a decay of the optically generated valley polarization. The impurityassisted exchange term X leads to a carrier transfer between the resonant states in the two valleys (A−A, B−B coupling). The intrinsic Coulomb-induced renormalization term Y couples A and B excitons of different valleys and the same spin. It does not cause carrier exchange between the valleys but leads to energetic shifts of the excitonic states in both valleys. (b) Origin of the dispersive profiles in the differential transmission spectra for same circular polarization (SCP) and opposite circular polarization (OCP). Due to the band gap renormalization in the valleys, the absorption of the pumped system shifts to lower photon energies. In addition, strong bleaching in the pumped K valley occurs due to phase space filling resulting in an asymmetric dispersive DTS curves.

OCP signal. In addition, the exchange coupling (X term) also known as electron−hole exchange interaction leads to a carrier transfer between the two valleys. However, this dipole−dipole coupling is relatively weak, since it is driven by the microscopic polarization, which quickly decays. Therefore, the observed redshifted dispersive shape and the intervalley coupling is dominantly caused by the Y term, as illustrated in Figure 3b, where the dashed lines show the DTS without the contribution of the X coupling. The reason for the more pronounced dispersive shape in OCP compared to SCP is the weaker renormalization of the Rabi frequency and the less efficient Pauli blocking because of the small amount of transferred charge carriers via X coupling to the unpumped K′ valley. The Y intervalley coupling causes a red-shifted absorption of the K′ excitons in the pumped case with respect to the unpumped case. Therefore, a positive differential transmission signal is found at the initial (unpumped) resonance position, while a negative signal is found at the red-shifted position, resulting in the observed strongly dispersive DTS curve (Figure 2b, right). 2948

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toward the original resonance of the unexcited system for delay times up to 30 ps. Within the first 10 ps the behavior is more complex. In the SCP measurement the negative component continuously shifts back to higher energies with increasing delay time (red dashed line in Figure 1b). This effect might be due to a decreasing charge carrier density resulting from exciton recombination and intervalley scattering from K to K′. In the OCP measurement intervalley scattering first leads to an increasing charge carrier density in K′. As a consequence, the redshift due to band gap renormalization increases, because now excitons populate the K′ valley. This leads to a redshift of the negative component in the OCP measurement within the first few picoseconds (blue dashed line Figure 1b). After relaxation of the valley polarization the negative component also shows a redshift as in SCP, because of exciton recombination. In conclusion, we have demonstrated that the K and K′ valley of atomically thin WS2 are strongly coupled by the Coulomb interaction.The off-resonant A−B excitonic coupling in different valleys is due to the intrinsic Coulomb-induced interaction (Y term). Additionally, the resonant A−A and B−B coupling of excitons in different valleys occurs via the disorder-induced electron−hole exchange interaction (X term). In the presented experiment, the intrinsic intervalley coupling (Y term) is found to be the dominant mechanism giving rise to the strongly dispersive OCP signal. The valley-selective optical excitation with circularly polarized light leads to a band gap renormalization and a red-shift of excitonic resonances both of the pumped K and unpumped K′ valley. In differential transmission, this spectral shift causes a prominent dispersive signal when probing the K′ valley. Since the occupation of the unpumped valley remains relatively small, the photoluminescence from the K′ valley is weak. This effect explains the high valley polarization degrees found in PL measurements, while at the same time both the SCP and OCP differential transmission signals are strong. The presented results are also applicable to other atomically thin TMDC materials, such as MoS2. Indeed, the intervalley coupling is expected to be the strongest for TMDCs with a small spin−orbit coupling, because in this case the intrinsic A−B coupling is most efficient.

Figure 3. (a) Experimental and (b) microsopically calculated normalized differential transmission spectra at zero delay time for different pump fluences/exciton densities. Pumping and probing the same valley (SCP, red) causes a pronounced absorption bleaching resulting in a positive signal. In contrast, probing the other valley (OCP, blue) leads to a dispersive signal including negative contributions due to the Coulomb-induced intervalley coupling. The dashed lines show calculations without the X term.

In both valleys, the change in the energy renormalization responsible for the dispersive DTS shape depends on the number of charge carriers pumped into the K valley. Hence, we observe a stronger shift to lower energies with increasing pump fluence. This effect is clearly visible in Figure 3, highlighted by the vertical dashed lines. A direct comparison between the measured and calculated spectra shows a good qualitative agreement of the main spectral features. At low pump fluences the theoretical and experimental OCP curves have nearly the same relative amplitude for the positive and negative DTS component due to the above-discussed small change in Pauli blocking. However, at higher pump fluences the negative component of the OCP signal becomes smaller compared to the positive component, which is due to a stronger influence of the X term. It creates a population of A excitons in the unpumped K′ valley and thereby leads to a stronger bleaching in the OCP measurements. This effect is evident by comparing the calculations with (solid lines) and without X term (dashed lines) in Figure 3, where the dispersive profile in the OCP curve remains symmetrical for higher pump fluences, if only the Y term is considered. After having explained the physical mechanisms at zero time delay, we now turn to the temporal evolution of the DTS features, shown in Figure 1. Since the band gap renormalization in both valleys is stronger with a higher charge carrier density in the K valley, the resulting red shift becomes smaller at larger delay times due to the loss of excitons by relaxation processes. This effect can be seen in Figure 1a and b, where the negative component of all differential transmission spectra shifts back

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.nanolett.5b04733. Sample characterization, details of the pump−probe setup, population dynamics, details of theoretical modeling (PDF)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Fax: +49 (0) 251 83-36414. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We gratefully acknowledge financial support from the Deutsche Forschungsgemeinschaft (SFB 787, SFB 951, SPP 1391, and FOR 1493) and the EU Graphene Flagship (contract no. CNECT-ICT-604391). We also thank M. Richter (TU Berlin) 2949

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(31) Tonndorf, P.; Schmidt, R.; Bottger, P.; Zhang, X.; Borner, J.; Liebig, A.; Albrecht, M.; Kloc, C.; Gordan, O.; Zahn, D. R. T.; de Vasconcellos, S. M.; Bratschitsch, R. Opt. Express 2013, 21, 4908− 4916. (32) Jin, W.; Yeh, P.-C.; Zaki, N.; Zhang, D.; Sadowski, J. T.; AlMahboob, A.; van der Zande, A. M.; Chenet, D. A.; Dadap, J. I.; Herman, I. P.; Sutter, P.; Hone, J.; Osgood, R. M. Phys. Rev. Lett. 2013, 111, 106801. (33) Zhao, Y.; Luo, X.; Li, H.; Zhang, J.; Araujo, P. T.; Gan, C. K.; Wu, J.; Zhang, H.; Quek, S. Y.; Dresselhaus, M. S.; Xiong, Q. Nano Lett. 2013, 13, 1007−1015. (34) Mak, K. F.; He, K.; Lee, C.; Lee, G. H.; Hone, J.; Heinz, T. F.; Shan, J. Nat. Mater. 2012, 12, 207−211. (35) Haug, H.; Koch, S. W. Quantum Theory of the Optical and Electronic Properties of Semiconductors, 5th ed.; World Scientific Publishing Co. Pre. Ltd., 2004. (36) Malic, E.; Knorr, A. Graphene and Carbon Nanotubes: Ultrafast Relaxation Dynamics and Optics, 1st ed.; Wiley-VCH: Berlin, 2013. (37) Lindberg, M.; Koch, S. W. Phys. Rev. B: Condens. Matter Mater. Phys. 1988, 38, 3342. (38) Knorr, A.; Hughes, S.; Stroucken, T.; Koch, S. W. Chem. Phys. 1996, 210, 27. (39) Kira, M.; Koch, S. W. Progress in Quantum Electronics; IEEE Journal of Quantum Electronics, 2006; Vol. 30. (40) Berghäuser, G.; Malic, E. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 89, 125309. (41) Moody, G.; Kavir Dass, C.; Hao, K.; Chen, C.-H.; Li, L.-J.; Singh, A.; Tran, K.; Clark, G.; Xu, X.; Berghäuser, G.; Malic, E.; Knorr, A.; Li, X. Nat. Commun. 2015, 6, 8315. (42) Arora, A.; Koperski, M.; Nogajewski, K.; Marcus, J.; Faugeras, C.; Potemski, M. Nanoscale 2015, 7, 10421−10429. (43) Selig, M.; Berghaeuser, G.; Malic, E.; Knorr, A. Submitted 2016. (44) Keldysh, L. V. JETP Lett. 1979, 29, 658−661. (45) Cudazzo, P.; Tokatly, I. V.; Rubio, A. Phys. Rev. B: Condens. Matter Mater. Phys. 2011, 84, 085406. (46) Berkelbach, T. C.; Hybertsen, M. S.; Reichman, D. R. Phys. Rev. B: Condens. Matter Mater. Phys. 2013, 88, 045318. (47) Berghäuser, G.; Malic, E. Phys. Rev. B: Condens. Matter Mater. Phys. 2014, 89, 125309. (48) Sercombe, D.; Schwarz, S.; Pozo-Zamudio, O. D.; Liu, F.; Robinson, B. J.; Chekhovich, E. a.; Tartakovskii, I. I.; Kolosov, O.; Tartakovskii, A. I. Sci. Rep. 2013, 3, 3489.

for helpful discussions on the implementation of the carrierimpurity interaction.

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DOI: 10.1021/acs.nanolett.5b04733 Nano Lett. 2016, 16, 2945−2950