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Efficient Excitonic Photoluminescence in Direct and Indirect Band Gap Monolayer MoS2 Alexander Steinhoff, Ji-Hee Kim, Frank Jahnke, Malte Rösner, Deok-Soo Kim, Chanwoo Lee, Gang Hee Han, Mun Seok Jeong, Tim Oliver Wehling, and Christopher Gies Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.5b02719 • Publication Date (Web): 31 Aug 2015 Downloaded from http://pubs.acs.org on September 1, 2015
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Nano Letters
Efficient Excitonic Photoluminescence in Direct and Indirect Band Gap Monolayer MoS2 A. Steinhoff,† J.-H. Kim,‡ F. Jahnke,† M. R¨osner,†,¶ D.-S. Kim,§ C. Lee,§ G.H. Han,‡ M. S. Jeong,‡,§ T.O. Wehling,†,¶ and C. Gies∗,† Institute for Theoretical Physics, University of Bremen, Germany, Center for Integrated Nanostructure Physics, Institute for Basic Science, Republic of Korea, Bremen Center for Computational Materials Science, and Department of Energy Science, Sungkyunkwan University, Republic of Korea E-mail:
[email protected]
Abstract
distribution of carriers predominantly in the Kvalleys, which leads to strong emission from the A-exciton transition and a visible B-peak even if the band-gap is indirect. For above-bandgap excitation, we predict a strongly reduced emission intensity at comparable carrier densities and the absence of B-exciton emission. The results agree well with PL measurements performed on monolayer MoS2 at excitation wavelengths of 405 nm (above) and 532 nm (below the band gap). Keywords: photoluminescence, MoS2 , transition metal dichalcogenide, strain engineering, many-body effects, 2D materials The optical properties of the single- and few layer transition metal dichalcogenides (TMDs) MoS2 , MoSe2 , WS2 and WSe2 have drawn considerable attention since first reports on efficient photoluminescence (PL). 1–3 Early studies of these new materials comprised the determination of the single-particle band structure and band gap, 4–6 and the optical transition energy of the exciton (sometimes refered to as “optical gap”), 7 which is the single-particle band gap minus the binding energy. More recently, the focus has shifted towards investigations of more intricate many-body effects, such as the appearance of higher excited bound states, 8 carrier dynamics, 9,10 line shifts 11 and bleaching of boundstate resonances 10,12 in the presence of excited
We discuss the photoluminescence (PL) of semiconducting transition metal dichalcogenides on the basis of experiments and a microscopic theory. The latter connects ab-initio calculations of the single-particle states and Coulomb matrix elements with a many-body description of optical emission spectra. For monolayer MoS2 , we study the PL efficiency at the excitonic A and B transitions in terms of carrier populations in the band structure and provide a quantitative comparison to an (In)GaAs quantum well-structure. Suppression and enhancement of PL under biaxial strain is quantified in terms of changes in the local extrema of the conduction and valence bands. The large exciton binding energy in MoS2 enables two distinctly different excitation methods: above-band-gap excitation, and quasi-resonant excitation of excitonic resonances below the single-particle band gap. The latter case creates a non-equilibrium ∗
To whom correspondence should be addressed Institut f¨ ur Theoretische Physik, Universit¨at Bremen, P.O. Box 330 440, 28334 Bremen, Germany ‡ Center for Integrated Nanostructure Physics, Institute for Basic Science (IBS), Suwon 440-746, Republic of Korea ¶ Bremen Center for Computational Materials Science, Universit¨ at Bremen, 28334 Bremen, Germany § Department of Energy Science, Sungkyunkwan University, Suwon 440-746, Republic of Korea †
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carriers, as well as optical gain. 10 Various manybody effects are similarly affected by modifications of the band structure, and manipulations of strain and dielectric environment have been suggested to tailor optical properties (“strain engineering”). 13–16 In this Letter, results of a microscopic theory are compared with experiments and with calculations for standard (In)GaAs quantum wells (QWs). While absorption spectra reflect the excitonic landscape in the band structure by probing the resonances with a weak light field, photoluminescence (PL) originates from the recombination of carriers populating excited states. Understanding luminescence therefore relies on the combined knowledge of band structure, excitonic effects induced by the Coulomb interaction, dipole coupling of the thin-layer material to the light field, and carrier-scattering processes. The latter govern the population dynamics following optical or electrical excitation and also determine the homogeneous linewidth. Experiments and device engineering are currently progressing at a faster pace than theoretical modeling of these systems. Still, fundamental aspects of something as simple as excitonic PL from monolayer TMDs are not fully understood in terms of many-body effects and excitation dynamics. We aim at closing this gap by presenting results from a microscopic theory, taking into account the materialrealistic band structure and dielectrically screenend Coulomb interaction on an ab-initio G0 W0 basis, carrier-density-dependent screening and renormalization effects, the impact of straininduced changes of the band-structure, and its population with excited carriers. The results are supported by experimental investigations of PL, where excitation is performed within and above the single-particle band gap. Both cases fundamentally differ regarding the carrier population that is created in the band structure. To quantify the possible PL efficiency of monolayer MoS2 in its pristine, freestanding form apart from defect-assisted processes, we provide a comparison to the PL efficiency of an (In)GaAs QW, representing a well-established and frequently used semiconductor material in optoelectronic devices. It is shown that the in-
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tegrated PL intensity of unstrained monolayer MoS2 can reach values comparable to those of the QW in a regime of typical carrier densities, or even outreach the PL intensity under tensile strain, for which we predict an emission enhancement by an order of magnitude. From a theoretical viewpoint, in the past the rich excitonic physics of MoS2 has solely been investigated by means of linear absorption spectra. GW calculations combined with a solution of the Bethe-Salpeter equation (BSE) have been successfully used to explain features seen in absorption measurements, while at the same time they treat only ground-state properties. 4,5,17,18 In the presence of excited carriers, many-particle renormalizations and Coulomb screening have been predicted 19 to cause a red shift and bleaching of the A and B peaks (but not the C-peak) in agreement with recent experiments. 10–12,20 Our theoretical modeling of PL is based on a two-step approach: First, the quasiparticle band structure and dielectrically screened Coulomb matrix elements are obtained from an ab-initio G0 W0 calculation. These results serve as input for many-body calculations of the interaction of excited carriers with the quantized electromagnetic field. Field quantization naturally introduces spontaneous emission— the origin of PL. The manybody emission properties are described by semiconductor luminescence equations (SLE), which are an established method in semiconductor physics. On various levels of approximation, they systematically account for excitonic effects between pairs of carriers (excitons), three charge carriers (trions), pairs of excitons (biexcitons) and so on. We present three main results: In Sec. I, emission intensity and line shifts due to many-body effects are related to carrier populations and compared quantitatively to an (In)GaAs QW system. We predict PL quenching at elevated carrier densities, caused by the accumulation of carriers in the local conduction-band minimum at the Σ point. In Sec. II, the straindependence of PL is investigated. In agreement with a recent experiment, strongly reduced emission is observed for both tensile and compressive strain, but we also predict a regime
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I. Excitonic Photoluminescence from Monolayer TMD systems. PL arises from the spontaneous recombination of electrons and holes under the influence of the Coulomb interaction, following optical excitation or electrical injection of carriers into the band structure, and taking place at the Coulomb-renormalized energies, such as the resonances of the exciton, trion, biexciton and so on. Many experiments have reported the occurrence of trions and biexcitons. 11,20–22 In the present work, we restrict ourselves to the description of excitonic PL, which correctly accounts for the binding energy in the recombination of electrons and holes. We point out that modifications of the band structure, caused by strain and by manyparticle renormalizations, affect both exciton and trion in the same way. In this section and in Sec. II, we assume that carriers in both spin-subsystems are excited and fast carrier scattering is able to redistribute them into a quasi-equilibrium before recombination sets in. Then, electrons and holes reside in their respective bands across the BZ with separate electron and hole chemical potentials. We expect this situation to be a possible scenario after aboveband-gap excitation or carrier injection, 23 see Sec. III for a detailed explanation. Typical PL spectra are shown in the inset to Fig. 1. In contrast to the rich excitonic signatures seen in the absorption, 1–3,5,13,17,19 only A-peak emission occurs in PL. This is in excellent agreement with the PL and absorption measurements reported in Ref. 1. The absence of the higher-lying B peak even at elevated carrier densities results from the absence of holes in the lower-valence-band maximum at K. This is a consequence of the high density of states in the K-valley and the large valence-band spinorbit splitting of 150 meV, due to which most carriers can, in equilibrium, be accommodated in the highest-lying valence band. In the left side of Fig. 1 we present as functions of carrier density theoretical results for integrated PL intensity emitted vertically from a freestanding monolayer of MoS2 and the spectral shift of the A-peak due to many-body effects. A direct comparison with the corresponding results for a 4 nm InGaAs QW embedded
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Figure 1: Left: Calculated integrated PL intensity (top) and spectral shift of the A-peak emission (bottom) for monolayer MoS2 as function of excited-carrier density. Blue curves correspond to the unstrained monolayer (lattice constant 3.18˚ A), red (3.16˚ A) and green (3.20˚ A) curves represent results under compressive and tensile strain, respectively, and are discussed in Sec. II. The inset shows typical PL spectra in a.u. for a carrier density of 1013 cm−2 following above-band-gap excitation. Right: Comparable data obtained for typical carrier densities in a 4 nm (In)GaAs QW. Emission takes place into the GaAs material, and the oscillator strength is reduced by 1/εr , which follows from Eq. (3). PL intensity is given in photons per µs emitted from 1µm2 of the sample in a solid angle with 1◦ opening vertical to the thin layer.
of strain-induced emission enhancement by an order of magnitude. In Sec. III theoretical and experimental results for two possible excitation scenarios are presented, namely quasi-resonant excitation of excitonic states, and non-resonant excitation above the single-particle band gap. We argue that this distinction plays a crucial role in TMD semiconductors and is enabled by the huge exciton binding energy inherent to these systems. Details regarding the specific application of the SLE to TMD materials are given in the end of this Letter, and the interested reader may wish to begin with this final part.
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faster than that of the K valley with increasing carrier density. As a consequence, at densities around 1012 cm−2 the band gap becomes indirect at Σ and only about 20% of the electrons accommodate the K valley. Further increase of the carrier density to 1013 cm−2 lowers the Σ valley even more (situation depicted in Fig. 2a), and most carriers end up there (Fig. 2b). Since excitonic PL originates from carriers located in the K valleys, carrier drain from K to Σ leads to the observed quenching. The described effect is particular to the band structure found in atomically thin TMD systems. In the (In)GaAs QW system, electrons and holes always populate the Γ valley, from where optical recombination takes place. Consequently, for the integrated QW PL, depicted in the right panel of Fig. 1, quenching is absent. The quantitative comparison also shows that similar photon flux is possible from unstrained monolayer MoS2 and the (In)GaAs QW reference system in the respective high-excitation regimes of both material systems. With increasing carrier density a significant red shift of the emission is observed (lower panel in Fig. 1). Excited carriers cause a reduction of the binding energy due to plasma screening and phase space filling and energy shifts renormalize the band gap. The observed red shift is a combination of these effects. The same behavior has been predicted 19 and observed 10,11 for absorption spectra in the presence of excited carriers. We point out that the used screened-exchange and Coulomb hole (SX-CH) approximation is known to overestimate line shifts at high densities, which should be kept in mind when comparing our results to experiments. This is also evident from the QW calculations that exhibit a sizable line shift at densities above 1011 cm−2 (Fig. 1), which has been shown to be absent if correlations are treated beyond the SX-CH approximation. 26 Independent and possibly in addition to the discussed line shift is the transfer of oscillator strength from neutral to charged excitons that, in the presence of strong line broadening, also leads to an apparent line shift, as reported e.g. in Ref. 20.
Figure 2: a) and c) Illustration of the carrier population (red, to scale) in the band structure (black curve) around the conductionband K and Σ points at a carrier density of 1013 cm−2 . For the unstrained case (top panel), the Σ-valley is shifted below the K-valley due to many-particle renormalizations and draws carriers from the K valley. Tensile strain (bottom panel) lowers the K valley significantly, so that a larger fraction of the carrier population resides there even at high densities. b) and d) Fraction of the carrier population in the K and K0 (labeled as K), as well as Σ and Σ0 (labeled as Σ) valleys as function of total carrier density for the unstrained (top) and tensile strained (bottom) case. The correspondence of the figures on the left side in marked by the stars.
in GaAs is provided in the right part of Fig. 1 and may serve as a benchmark for the TMD system. For the (In)GaAs material system the effective-mass approximation is used, which is known to be justified. 24,25 The integrated MoS2 PL intensity (blue curve in the top panel for the unstrained monolayer) follows a linear increase with carrier density up to 3 × 1012 cm−2 . At higher densities, a much slower increase of the PL intensity is predicted. This behaviour can be traced back to the distribution of electrons between the K and Σ valleys in the conduction band shown in Fig. 2a) and b). While the band gap is known to be direct at the K point in the ground state, many-particle renormalizations lower the position of the Σ valley
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sile strain, the integrated PL intensity from the TMD shown in Fig. 1 is enhanced by one order of magnitude (green curve) over the unstrained case and exceeds that of the QW system. Compressive strain acts the opposite way and lowers the energy of the Σ valley, making the band gap indirect even in the absence of carrierdensity-induced band renormalizations. As a consequence, PL intensity is strongly reduced and even more susceptible to quenching than in the unstrained case (red curve in Fig. 1). Local band-structure modifications have been used to explain experimentally observed straindependence of the PL intensity in Ref. 13. In the experiment, uniaxial strain leads to a shift of the valence-band Γ valley until it resides at higher energies than the K valley. In this case, holes in the valence-band are drawn to the Γ point, which has a result similar to the abovediscussed loss of conduction-band carriers into the Σ valley. We find a comparable situation when biaxial tensile strain is increased to about 2% (Fig. 3c). The strong decrease of hole population at K consequently leads to a roll-over of the total PL intensity with increasing strain, which is shown in Fig. 3a). This quenching of PL intensity under strong tensile strain is in very good qualitative agreement with the experimental observation. Our prediction of a prior increase of intensity for smaller values of tensile strain is not observed in Ref. 13, which might be explained by the fact that the band structure of freestanding, unstrained MoS2 as used in our theory can differ from that of a sample in the laboratory and on a substrate. In the context of semiconducting TMDs, discussions in the literature often name the direct band gap as a strong advantage of the monolayer over bi- or multi-layers, for which the band gap is indirect. It is interesting to see that also in the monolayer, depending on the strain situation and substrate effects, the band gap is likely to be indirect at elevated excitation levels, limiting the full potential of PL yield especially when off-resonant excitation is used, such as electrical carrier injection. For the quasiresonant excitation of excitonic transitions the situation will be distinctly different.
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Figure 3: a) Integrated PL intensity as function of biaxial strain applied to the freestanding MoS2 monolayer at a carrier density of 1013 cm−2 . b) Lowest conduction band in the vicinity of the K and Σ valleys for lattice constants of 3.16˚ A (red), 3.18˚ A (blue), and 3.20˚ A (green), and c) highest valence band in the vicinity of the K and Γ valleys for lattice constants of 3.16˚ A (red), 3.18˚ A (blue), and 3.25˚ A (green) at a carrier density of 1013 cm−2 . Bands are to scale. The dashed lines mark the respective positions of the extremum at K.
II. Strain-induced PL Enhancement and Suppression. The band structure sensitively depends on strain. It has been shown that strain can have two distinct effects: (i) a shift of the emission wavelengths and (ii) a possible reduction of the oscillator strength. 13 The impact of tensile and compressive strain on the bandstructure extrema in the lowest conduction- and highest valence band is shown in Fig. 3b) and c). In the conduction band, tensile strain increases the offset between the K and Σ valleys, thereby making the band gap at K “more direct”. As a consequence, the K-valley can accommodate higher carrier densities before the Σ valley gets populated, which counteracts the above-discussed quenching of the PL. This is quantitatively demonstrated in Fig. 2d): At a density of 1012 cm−2 , carriers nearly exclusively populate the K valley. At higher densities, the Σ valley is lowered, but the fraction of carriers in the K valleys remains significantly higher than in the unstrained case. For 0.6% ten-
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III. PL following Quasi-Resonant and Above-Band-Gap Excitation in TMDs. Owed to the large binding energy of the excitonic states, there is a sizable energetic window between the lowest excitonic transition (the Aexciton) and the true single-particle band gap of about 2.5 eV as predicted 4,5,17,18 from GW calculations for MoS2 . Above this gap no bound states exist at the K point in the Brillouin zone (BZ). As a result, excitation at energies within and above this window leads to different carrier populations and emission properties in the successive PL. Above band-gap excitation creates an electron-hole plasma in energetically higherlying k-states in the BZ. Recent experiments have demonstrated fast carrier dynamics on ps timescales due to efficient carrier-carrier and carrier-phonon interaction. 9,27–29 Fast relaxation redistributes carriers into a quasiequilibrium, in which electrons (holes) occupy the respective minima (maxima) of the band structure. As it is known from semiconductor heterostructures, the resulting excited-state population will exist as a mixture of electronhole plasma and bound excitons (see the discussion in the Supporting Information). 30 Quasi-resonant excitation below the band gap is possible only due to the existence of excitonic states. The spectral overlap between the excitonic bound-state resonances and the optical excitation allows the coherent optical field to create an appreciable polarization in the system. As described by the dynamics on an optical Bloch sphere 31 (for each k point), this polarization drives coherent carrier populations that can, due to dephasing, become incoherent and remain in the system even after the excitation is over. The result is a population of carriers at the respective k-points in the BZ where the bound-state wave functions reside. For the A and B excitons, which exhibit resonances between 1.9 and 2.0 eV, and also the corresponding excited states with resonances roughly 0.3 eV higher, 5,8,19 this is mainly (but not exclusively) in the K valleys of the BZ. 5,19 A prediction of the actual carrier distribution in an experiment under continuous wave (cw) optical excitation requires knowledge of
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the carrier scattering processes and rates. Research on this complex topic has only recently begun for TMD systems, 9,27–29 and a theoretical model for non-equilibrium carrier dynamics is not yet available. Such calculations will also provide insight in how far two-particle correlations between electrons and holes will form bound excitons, and how persistent these correlations are with respect to thermalization due to scattering processes. To capture the main effects of the different excitation scenarios, we consider two limiting cases: For aboveband-gap excitation, we assume that carriers in both spin-subsystems are excited and fast carrier scattering is able to redistribute them into quasi-equilibrium before recombination sets in. This situation has been used in Secs. I and II. To model quasi-resonant excitation below the single-particle band gap, we use a two-step process. First, semiconductor Bloch equations 19,32 are solved in time to describe the coherent carrier generation from a circularly polarized excitation pulse 60 meV above the A-exciton energy with a spectral full width at half maximum of 5 meV. For these parameters, the excitation pulse overlaps both with the A- and, to a lesser extent, the B-exciton resonances. In a second step, the PL spectrum is obtained by solving the SLE under the assumption that the resulting non-equilibrium carrier distributions persist during the recombination process. Time-integrated PL spectroscopy has been performed on chemical-vapor-deposition (CVD) grown MoS2 on SiO2 (300 nm) using a NT-MDT (NTEGRA-SPECTRA) confocal system at room temperature. Excitation has been performed using linearly polarized cw lasers with four different wave lengths, 405 nm (3.1 eV), 473 nm (2.6 eV), 532 nm (2.3 eV), and 633 nm (1.96 eV). The laser spot size was about 0.5 µm with 1 mW power, and an average number of about 1.4 × 1013 cm−3 photons could be obtained from the sample at all excitation wavelengths. We collected the PL spectra using a thermoelectrically cooled CCD camera, and the 150 grooves grating was used. In the top panel of Fig. 4 we compare results for above-band-gap and quasi-resonant excitation obtained from experiment and theory.
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1 a)
sured PL spectra (Fig. 4a) are compared for excitation wavelengths of 405 nm (corresponding to above-band-gap excitation at 3.1 eV) and 532 nm (quasi-resonant excitation at 2.3 eV), at which presumably higher resonances of the A or B exciton are found. 8 Instead of explicitly treating the carrier relaxation following excitation with larger excess energy, we consider pumping spectrally between the ground states of A and B excitons. This is expected to lead to a comparable net result, as according to theoretical predictions, the wave functions of the higher bound states are similar (even if somewhat more localized in k-space) to those of the ground-state bound states. 19 The most prominent feature in both calculated and measured spectra is, besides the dominant emission from the A-exciton resonance, the presence of visible B-exciton emission for excitation below the band gap, and its absence for above-band-gap excitation. For quasiresonant excitation, the theoretical result is explained in terms of the coherent excitation at the B-exciton resonance that results in holes occupying the lower-lying conduction band at K, from where B-exciton emission is possible. From an experimental point of view, the observation of B-exciton emission is in agreement with many results reported in the literature, as most often excitation is performed at wavelengths below the single-particle band gap. 2,3,12,15,23 Additional experimental results shown in Fig. 4c) demonstrate that the PL emission is strongly dependent on the excitation wave length if excitation takes place below the band gap. At 633 nm, the excitation energy lies between the A and B resonances, with a much stronger overlap with the B-exciton resonance. The resulting PL from the B peak is visible up to the wavelength of the exciting laser and of comparable intensity than the A-peak emission. A similar result is seen for excitation at 473 nm. New insight is provided by the measurement performed at 405 nm excitation. The absence of B-peak emission from the otherwise identical material system at all excitation levels (Fig. 4d) supports our explanation in terms of the formation of a quasi-equilibrium carrier distribution
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Theoretical results (Fig. 4b) are calculated for comparable carrier densities and according to the excitation schemes discussed above. Mea-
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instead of a non-equilibrium population of carriers in the K valleys. We emphasize that reality may deviate from these simplified pictures due to the discussed dynamics of carrier scattering and the formation of carrier correlations. Nevertheless, the presented results offer insight on a microscopic level that may aid further understanding of more complex topics, such as the emission efficiency of electrically driven integrated devices. The given interpretation allows for a possible explanation of experimental results for electroluminescence (EL) presented in the literature. 21,23 We expect electrical injection of electrons and holes to be comparable to optical above-band-gap excitation, as carriers are created high in the bands, so that resulting carrier distributions will more closely correspond to quasi-equilibrium distributions, for which Bexciton emission is suppressed. EL and PL have been measured side by side by Sundaram and co-workers in Ref. 23. Under optical excitation below the band gap, an emission feature is evident in the PL spectrum that likely corresponds to the B peak, and which is absent under electrical excitation. While the electrical excitation is weak in the experiment, this result may further support the above discussion on the difference between excitation within and above the single-particle band gap. Due to the increased population of carriers in the K valleys, we predict a significantly higher emission intensity in case of quasi-resonant excitation. The implications are particularly relevant in the context of indirect semiconductor materials and may explain why significant PL is seen from the direct transition in the bilayer, and even in up to six layers, 2,3,12,33 for which the band gap is known to be indirect. 34,35 A quantitative comparison of the emission intensity between above-band-gap and quasi-resonant excitation is provided in the Supporting Information for compressively strained monolayer MoS2 which is an indirect semiconductor already in the ground state.
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culate PL properties and give details how it is applied to the TMD material system. PL is a quantum-mechanical process triggered by spontaneous emission, which naturally arises from the quantization of the electromagnetic field. The equations describing the coupling of electronic transitions in a semiconductor to a continuum of modes of the quantized electromagnetic field are known as semiconductor luminescence equations (SLE) and are an established method to model emission properties of QW and quantum-dot systems. 32,36–38 Furthermore, they are the foundation of modern laser theories that are able to access statistical properties of the light emission. 39,40 The PL spectrum I(ω) is defined by I(ω)dω being the rate of photon emission into a certain frequency range. In the limit of high frequency resolution of a detector, the emission rate is given by the time derivative of the total population of all free-space photon modes ξ of the quantized field that have frequencies in the respective range: 38 I(ω)dω =
X ξ,ωξ ∈[ω,ω+dω]
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Here, the mode label ξ contains both the wave vector q and the polarization σ of the photons. The Hamiltonian of the coupled carrier0 0 photon system H = Hcarr + Hph + HD + 0 HCoul consists of the electronic part Hcarr , the 0 free electromagnetic-field contribution Hph , the Hamiltonian of the quantized light-matter interaction HD in Dipole approximation, and the Coulomb-interaction Hamiltonian HCoul that are spelled out in the Supporting Information. The quasi-particle band structure and the Coulomb matrix elements that enter are obtained on the basis of G0 W0 calculations for monolayer MoS2 . Both the band structure and the Coulomb matrix elements are projected onto proper Wannier orbitals, which are predominantly composed of Mo-dz2 , -dxy and -dx2 y2 orbitals but also carry some Sp weight. 41 The band structure is obtained with the PAW (GGA-PBE) 42 method as implemented in the Vienna Ab initio Simulation Package (VASP), 43,44 whereas the Coulomb ma-
Semiconductor Luminescence Equations for TMD Systems. In this last section, we introduce the microscopic model used to cal-
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trix elements are calculated with the SPEX 45 code involving Kohn-Sham eigenstates calculated within the FLAPW method as implemented in the FLEUR 46 code. To overcome artificial self-screening effects of adjacent slabs in the periodic unit cell we extrapolate the G0 W0 band gap as well as the Coulomb matrix elements in dependence of the vacuum height (the separation of neighbouring MoS2 sheets). For more details on the ab-initio method and the calculation of dipole matrix elements we refer to Ref. 19. Based on the total Hamiltonian, Heisenberg’s equations of motion are used to derive a hierarchy of dynamical equations for quantummechanical expectation values, and the SLE typically refer to the lowest-order truncation of this hierarchy. The SLE comprise the coupled equations for the photon-population dynamics i¯h
bution, which is in Hartree-Fock approximation proportional to the product of electron and hole populations and the dipole matrix elements (line 4 of Eq. (2b)). The described transition does not take place at free carriers energies, but the Coulomb interaction renormalizes the band structure via εee,h = εe,h + ΣHF, e,h , and introduces excitonic effects (line 3) that lead to the exciton binding energy and the observed emission frequencies in the spectrum. We go beyond the Hartree-Fock level by including correlations that give cor(t) to the spontaneous-emission rections Ωscatt,he k,ξ term 30,36 contained in the last line, which we discuss in the Supporting Information. Moreover, we introduced phenomenological dephasing in Eq. (2b) by adding an imaginary part iγ to the homogenous term, which damps the oscillations of the photon-assisted polarizations. Contributions from photon-assisted polarizations over the whole BZ enter the photonpopulation change in Eq. (2a). We limit ourselves to direct interband transitions between the two highest valence and the two lowest conduction bands. In dipole approximation, the corresponding light-matter coupling coefficient eh gk,ξ is given by
E D X d eh ∗ (gk,ξ ) b†ξ ck,h ck,e (t), nξ (t) = 2iRe dt k,h,e (2a)
which are driven by the photon-assisted transition amplitudes i¯h
E dD † bξ ck,h ck,e (t) = dt D E † e h (e εk + εek − h ¯ ωξ − iγ) bξ ck,h ck,e (t)
r eh gk,ξ
he,OD + Vk,ξ (t)(1 − fke (t) − fkh (t))
=
h ¯ ωξ ek d · eσ (q)Uq (r) hk (3) 2ε0 εr V
with the photon mode function Uq (r), mode volume V , and the photon polarization vector eσ (q). The latter can assume two orthogonal directions perpendicular to the propagation direction given by q. The dielectric constant of the medium into which the emission takes place directly enters Eq. (3). For the embedded (In)GaAs QW, a value of εr = 12.5 is used. For the freestanding monolayer TMD material, εr = 1. Many-particle renormalizations are included in the SX-CH approximation, 19,31 which leads to the following expression for the off-diagonal interaction: D E X he,OD † eh0 he0 Vk,ξ (t) = Wkk 0 kk0 bξ ck0 ,h0 ck0 ,e0 (t)/A
eh e + igk,ξ fk (t)fkh (t)
+ Ωscatt,he (t). k,ξ (2b) The PL spectrum (Eq. (1)) can then be obtained from the time evolution of Eqs. (2). Here, b†ξ is the creation operator of a photon in mode ξ, c†k,{e,h} are field operators that create an electron/hole with momentum k, ωξ is the photon frequency, and fke,h (t) are the single-particle-state occupancies as shown in Fig. 2. The photon-assisted polarization describes a carrier transition from the conduction to the valence band via the emission of a photon. This quantum-mechanical process is driven by the spontaneous emission contri-
k0 ,h0 ,e0
(4)
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as well as to modified band-structure renormalizations, see also the Supporting Information. We note here that the Coulomb matrix elements eh0 he0 Wkk 0 kk0 include material specific background screening of the MoS2 monolayer and plasma screening due to excited carriers via a longitudinal dielectric function, as discussed in Ref. 19. Thereby we assume that the carrier density fully contributes to the screening as uncorrelated carriers, while a finite ratio of correlated carriers would lead to weaker screening due to the charge neutrality of excitons. At this level of approximation, it is not possible to determine the degree of correlation of the excited carriers, which is precisely determined by the interband correlations discussed in the Supporting Information. We point out that while weaker screening would modify the density-dependent spectral shifts of emission lines, it would not change the main results of this work. More details on the derivation and evaluation of the equations as well as the origin of the spontaneous emission correction term Ωscatt,he (t) can be found in the Supporting Ink,ξ formation.
PL. Within the band gap, excitonic resonances are coherently driven and a non-equilibrium carrier distribution is created at the k points of the corresponding bound-state wave functions. The resulting PL is strongly dependent on the excitation wave length, which determines which excitonic resonances are driven. At the same time, it is less sensitive to the local band-structure landscape, and emission strongly exceeding that following above-bandgap or electrical excitation is expected especially in the presence of an indirect band gap, such as in multi-layer TMDs or compressively strained monolayer MoS2 . The microscopic many-body model provides insight into the underlying physical processes and the interplay of many-particle renormalization effects that determine the emission properties. We have evaluated the theory under the assumption of static carrier distributions. It will be important to address carrier dynamics, as well as the formation of correlations between carriers (formation of exciton populations, trions, biexcitons) and their impact on PL in future research.
Conclusion. The optical properties of semiconducting TMDs are strongly influenced by many-body effects. The presence of several extrema in close energetic vicinity (K and Σ in the conduction band, K and Γ in the valence band) manifests itself in a strongly density- and strain-dependent emission intensity yield. The direct band gap, for which monolayer MoS2 is well regarded, disappears due to band-gap renormalizations already at moderate excitation levels, which suppresses the emission especially under electrical or above-band-gap excitation. We predict PL enhancement by an order of magnitude for slight tensile strain of the freestanding monolayer, which counteracts the loss of the direct band gap. With that, emission intensity similar or even greater than that of a typical (In)GaAs quantum-well system can be expected for the pristine material. From theory and experiment we conclude that excitation with wave lengths above and within the single-particle band gap lead to fundamentally different carrier distributions and resulting
Associated Content. Supporting Information is available free of charge via the internet at http://pubs.acs.org. Acknowledgement This work has been supported by the Deutsche Forschungsgemeinschaft, the European Graphene Flagship, and the IBS-R011-D1 grant. The authors acknowledge resources for computational time at the HLRN (Hannover/Berlin). The authors declare no competing financial interest.
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