Abstract
When electronhole pairs are excited in a semiconductor, it is a priori not clear if they form a plasma of unbound fermionic particles or a gas of composite bosons called excitons. Usually, the exciton phase is associated with low temperatures. In atomically thin transition metal dichalcogenide semiconductors, excitons are particularly important even at room temperature due to strong Coulomb interaction and a large exciton density of states. Using stateoftheart manybody theory, we show that the thermodynamic fission–fusion balance of excitons and electronhole plasma can be efficiently tuned via the dielectric environment as well as charge carrier doping. We propose the observation of these effects by studying exciton satellites in photoemission and tunneling spectroscopy, which present direct solidstate counterparts of highenergy collider experiments on the induced fission of composite particles.
Introduction
The interplay of excitons and unbound electronhole pairs is at the heart of excitedsemiconductor physics. Due to exceptionally strong electronhole Coulomb interaction and a naturally high sensitivity to spectroscopic methods, atomically thin semiconductors from the class of transition metal dichalcogenides (TMDCs) are perfectly suited to study the fission of excitons. The latter present a prominent realization of composite bosons formed by fermionic constituents and therefore provide insight beyond the specific material class of twodimensional semiconductors. The prominent role of excitons in the optical properties of TMDCs suggests an interpretation of experimental results as well as theoretical prediction in terms of excitons rather than unbound electrons and holes^{1,2,3,4,5}. On the other hand, it is well known that, at a certain excitation density of electronhole pairs, the Mott transition is observed^{6,7,8}. Here a phase where excitons and unbound carriers can coexist evolves into a fully ionized electronhole plasma.
Since excitons are more or less neutral compound bosonic particles, manyparticle renormalization and screening effects in an exciton gas are very different from those in a plasma of unbound electrons and holes, which we refer to in the following as quasifree particles. For this reason, it is highly desirable to quantify the relative importance of excitonic and plasma effects over a wide range of electronhole excitation densities and to learn how it can be manipulated from the outside. It has already been suggested to tune exciton binding energies by electrical doping^{8} and some effort has been devoted to study the influence of dielectric screening on excitons in TMDC semiconductors^{9,10,11,12,13}. In the past, a powerful scheme has been developed to theoretically describe the balance between fission and fusion of excitons and quasifree particles, also termed ionization equilibrium^{14,15,16,17,18,19}, with applications to atomic plasmas and highly excited semiconductors. The scheme relies on the assumption of a quasiequilibrium between plasma and excitons being established before electronhole recombination sets in. This is supported by ultrafast equilibration due to efficient carrier–carrier^{20} and carrier–phonon interaction^{5} as well as exciton formation^{21,22} after optical excitation, see ref. ^{23} for a review.
Experimental verification of the ionization equilibrium has been achieved in GaAs quantum wells using THz spectroscopy to probe transitions between 1s and 2pexciton states^{23,24}. A similar technique in the midinfrared range has been applied recently to monolayer WSe_{2} ^{22}. Alternatively, the fractions of excitons and plasma can be determined from their contributions to photoluminescence spectra^{25} in combination with additional photoluminescence simulations.
Here we show that a phase largely dominated by excitons at elevated excitation densities and its abrupt transformation into an electronhole plasma at the Mott transition are found at room temperature in monolayer TMDC materials. At low densities, exciton fission due to entropy effects is predicted. We demonstrate that the thermodynamical balance between fission and fusion of excitons and quasifree particles can be directly manipulated by the choice of dielectric environment as well as charge carrier doping. We also suggest that new ways to quantify the fission–fusion balance are angularresolved photoemission spectroscopy (ARPES) and scanning tunneling spectroscopy (STS). The observation of excitons by these methods can at the same time be understood as fission of composite bosons, induced by incident photons or applied voltage. Photoemission spectroscopy also gives access to the extent of exciton wave functions via the structure of exciton satellites. To obtain quantitative results for the materials MX_{2} (M = W,Mo and X = S,Se), we build on the theory of ionization equilibrium, combining it for the first time with materialrealistic band structure and Coulomb matrix element calculations that enable us also to study the influence of the dielectric environment. Beyond frequencydependent plasma screening, we additionally include screening due to excitons. The latter is shown to be relevant although it has neither been discussed before in the context of ionization equilibrium nor for twodimensional materials.
Results
Spectral functions and exciton satellites
To examine the equilibrium properties of excited carriers in TMDCs, we use the quantumstatistical expression for the carrier density n _{ a } of the species a, which can be electrons or holes, as a function of temperature T and chemical potential μ _{ a } as a starting point:
f ^{a}(ω) denotes the Fermi distribution function depending on μ _{ a } and T, \({\cal A}\) is the crystal area and \(A_{{\bf{k}}\sigma }^a(\omega ) = 2i{\rm{Im}}G_{{\bf{k}}\sigma }^{{\rm{ret}},a}(\omega )\) is the spectral function of the singleparticle state k σa〉 related to the retarded singleparticle Green’s function
The selfenergy \({\rm{\Sigma }}_{{\bf{k}}\sigma }^{{\rm{ret}},a}(\omega )\) accounts for manyparticle effects giving rise to renormalizations of the singleparticle band structure \(\varepsilon _{{\bf{k}}\sigma }^a\) as well as contributions of bound states. For a given selfenergy, the inversion of Eq. (1) yields the chemical potential μ _{ a }(n _{ a }, T) for each species and therefore any thermodynamic property of the system in the grand canonical formulation. As we describe in detail in the “Methods” section, by using a Tmatrix selfenergy in screened ladder approximation and assuming small quasiparticle damping, we obtain a spectral function A ^{a}(ω) in the socalled extended quasiparticle approximation. It exhibits poles for quasifree and bound carriers as shown in Fig. 1. From the multiple valleys in the singleparticle band structure of electrons and holes (Fig. 1a), a rich spectrum of bound states emerges (Fig. 1b) which contains a variety of dark excitons with large total momentum Q besides the bright Kvalley excitons commonly referred to as A and B. The dark excitons, though playing a minor role in optical experiments, are essential to the description of the ionization equilibrium. Various bound states are reflected in the lowenergy satellites of the singleparticle spectral function. Excitonic contributions are expected to be observed in experiments that are sensitive to these spectral properties. In ARPES^{26}, momentumresolved images of the electron spectral function comparable to Fig. 1c are obtained, which are weighted with Fermi distribution functions that are defined by the chemical potential μ _{e} and temperature T. In quasiequilibrium, the measured intensity is given by
For fixed quasimomentum k, as shown in Fig. 1e, this is typically referred to as energy distribution curve. On the other hand, STS^{27} probes the local density of states and thus momentumaveraged spectral functions of electrons and holes, which are displayed in Fig. 1c, d. We therefore propose to use these wellestablished experimental techniques to spectrally distinguish between excitons and quasifree carriers. To this end, spectroscopy has to be combined with prior optical excitation of the semiconductor, requiring timeresolved experiments. Timeresolved ARPES with high temporal and spectral resolution has recently been applied to twodimensional materials, which prove to be highly sensitive to this method^{28,29,30}. In addition, atomically thin semiconductors, that are available in everimproving sample quality, have the advantage of a large spectral separation of excitonic and quasiparticle signatures, which is well above the available energy resolution. These modern developments open the possibility to quantify the degree of exciton ionization and to access the extent of exciton wave functions, which are encoded in the structure of the exciton satellites.
Ionization equilibrium and Mott transition
The spectral function in extended quasiparticle approximation is given by
where \({\rm{\Gamma }}_{{\bf{k}}\sigma }^a(\omega )\) and the renormalization factor \(Z_{{\bf{k}}\sigma }^a\) account for twoparticle states, as discussed in detail in the “Methods” section. According to Eq. (1), the spectral function can be used to separate the total electron and hole density (a = e, h),
into contributions from quasifree carriers and from carriers bound as excitons. The excitons, defined by twoparticle energies below the quasiparticle band gap, are approximately treated as bosons. Hence, the properties of the excited semiconductor at a given temperature and excitation density are described by the density of electrons n _{e}, the density of holes n _{h} and the density of excitons n _{X}. The degree of ionization of the excited carriers
will be established as a result of the ionization equilibrium between electrons, holes and excitons. While for optical excitation, equal densities of electrons and holes are generated, we distinguish here between electron and hole ionization to also include the effect of carrier doping, where electron and hole densities are different.
Using singleparticle band structures and boundstate spectra, which are determined as discussed in the “Methods” section, we solve Eq. (5) numerically to obtain the degree of ionization α _{ a } in various TMDC materials under different experimental conditions. The results are collected in Fig. 2 and exhibit the behaviour of the ionization degree as a function of the excitation density. There are different regimes of ionization to be observed. At high excitation densities between 3 × 10^{12} cm^{−2} and 1 × 10^{13} cm^{−2}, depending on experimental parameters, efficient screening and manyparticle renormalizations lead to a full ionization of excited carriers, which is known as Mott effect. At lower densities around n _{ a } = 1 × 10^{12} cm^{−2}, excitons dominate the physical properties of TMDCs for the parameters studied here due to the large excitonbinding energies and a density of states with dominant contributions from dark excitons. Bright excitons with very small momenta that are optically active make up only a tiny fraction of the total exciton density, as illustrated in Fig. 3. The density of bright excitons is smaller than the total exciton density by about five orders of magnitude in MoS_{2} and six orders of magnitude in WSe_{2} over the whole range of excitation densities below the Mott transition. Although only bright excitons directly recombine, excitons with larger momentum can relax via efficient exciton–phonon interaction^{5} and refill optically active states, thereby representing an efficient reservoir for bright excitons. In Fig. 3, we also provide the density of all intravalley excitons formed by carriers with equal spins in the K and K′ valleys. These excitons make up a much larger fraction of the total exciton density, while still most of the excitons feature electrons and holes with different spins and/or valleys.
As Fig. 2b shows, an efficient tuning knob for the degree of ionization is the dielectric screening due to the environment, which can change over a wide range depending on the experimental situation or device realization in which the TMDC monolayer is used. The reason is the strong impact of dielectric screening on the exciton binding energies. Typical examples for substrates are Borofloat (ε = 2), SiO_{2} (ε = 4) and sapphire (ε = 10). The dielectric constant of the environment on top of the monolayer is often given by the vacuum value. On the other hand, in devices the TMDC monolayer is usually fully encapsulated by dielectric material. As an example we consider a full dielectric enclosure with ε = 10, which might be either sapphire or additional layers of TMDC material in a vertical heterostructure whose main influence on the excitons is the dielectric screening^{31}. We find that the minimal degree of ionization can be tuned from below 0.1% (99.9% excitons) for weak dielectric screening to about 30% for strong screening, while the Mott density is lowered at the same time by about a factor of 3. The second important parameter, that is relevant for applications of TMDC monolayers, is the doping with additional carriers which might be either intrinsic or induced by external electric fields in a capacitor structure. Here the fractions of ionized electrons and holes, α _{e} and α _{h}, are discussed separately as the densities of the species are not equal anymore. We consider hole doping of WS_{2}, but similar results are expected in case of electron doping. According to Fig. 2e, f, even for weak doping the minority carriers are practically all bound as excitons below the Mott transition. On the other hand, for higher doping levels an increasing fraction of majority carriers exists as quasifree plasma due to missing partners for exciton formation. As a function of minoritycarrier density, the Mott transition is lowered by about the density of doped excess carriers. From this, we conclude that at doping levels above 10^{13} cm^{−2} neither dark nor bright excitons will exist in any case. This is facilitated by the experimental estimate for dopinginduced ionization at several 10^{13} cm ^{– 2} ^{8}. As Fig. 2c shows, the temperature is another crucial parameter, which can vary in experiments or devices due to heating of the active material under strong optical or electrical pumping. This effect has been used to explain the observed excitontoplasma ratio in monolayer WSe_{2} in ref. ^{22}. At room temperature and even at elevated temperatures up to 700 K, excitons clearly dominate below the Mott transition. At the same time, the Mott density slightly increases with temperature due to weaker renormalizations of the quasiparticle gap. It turns out that strain is no efficient tuning knob as both, bright and dark excitons contribute to the ionization equilibrium, although bright excitons are preferred in moderately tensilestrained TMDCs, see Supplementary Fig. 2c. A comparison of different TMDC materials shows that excitons are slightly more important in molybdenum than in tungstenbased TMDCs due to the larger binding energies, which leads to higher Mott densities.
When approaching the Mott density from the lowdensity side, manyparticle renormalizations, as given by Eq. (25) in the “Methods” section, become increasingly important. Exchange interaction and efficient screening due to free carriers as well as excitons reduce the quasiparticle band gap and the exciton binding energies. More and more excitons are ionized, which leads to an increase of efficient freecarrier screening and thereby to a selfamplification of the ionization effect until all excitons are dissociated into an electronhole plasma and the degree of ionization becomes α _{ a } = 1. Note that α _{ a } includes not only bright but also dark excitons with large total momenta for example between K and Σ valleys. Those excitons may have larger binding energies, as also discussed in ref. ^{32}, and they are slightly more stable against ionization than bright excitons visible in an optical experiment. Fig. 4 shows an illustration of the Mott effect in terms of the spectral functions in extended quasiparticle approximation, which contain both exciton and quasifreeparticle signatures. At very low excitation densities, the only spectral contribution stems from quasifree carriers at the band edge. With increasing density, the quasiparticle peak is shifted to lower energies due to manyparticle renormalizations. At the same time, spectral weight is transferred from the unbound quasiparticle to the boundstate peaks as exciton populations increase, see the explicit expression of the spectral function in Eq. (23). The appearence of several exciton satellites in the hole spectral function is due to different bound states involving electrons either in the K or Σvalleys, see Fig. 1b. The energetic position of a boundstate peak in the spectral function of carrier a is given by the difference of the corresponding boundstate energy E ^{ab}, which is an eigenenergy of the Bethe–Salpeter Eq. (16), and the energy of the second carrier b involved in the bound state. The bound resonance might therefore be interpreted as an effective ionization energy of the actual carrier a with respect to its energy in the quasiparticle band structure. Both the quasiparticle band structure and the effective ionization energies are observable in experiments that are sensitive to the singleparticle spectral function such as ARPES and STS. Despite the fact that the amplitudes of boundstate resonances in the spectral functions are relatively small, observables like the carrier density, Eq. (1), and the photoemission intensity, Eq. (3), involve weighting with a Fermi function that strongly favors the lowenergy resonances over the quasifree contribution. With increasing excitation density, quasiparticle and excitonic resonances approach each other until at the Mott density all excitons are ionized and only a quasiparticle peak of unbound carriers remains. Figure 5 shows the reduction of the quasiparticle gap until the Mott transition appears around n _{ a } = 8 × 10^{12} cm^{−2}.
An alternative picture of the interacting electrons and holes, which is consistent with the extended quasiparticle approximation, is the socalled “chemical picture”, in which excitons are considered as a new particle species besides electrons and holes^{18,19}. They are characterized by a chemical potential
with boundstate energies E _{ ν } that are given by the relative motion of electron and hole, and an ideal Bose distribution function. In the chemical picture, solving Eq. (5) corresponds to an adaption of the chemical potentials of the different particle species, namely electrons, holes and excitons, as in a chemical reaction. These considerations are consistent with the theory based on spectral functions, that we use to obtain all numerical results presented in this paper. Only for the purpose of illustration, we simplify the theory considering the nondegenerate case (\({f^a}(E_{{\bf{k}}\sigma }^a) \ll 1\)), a single bandstructure valley for electrons and holes each and momentumindependent bandstructure renormalizations. Then a Saha equation can be formulated that determines the degree of ionization:
In analogy to the usual mass action law, \(I_\nu ^{{\rm{eff}}} = {\rm{\Delta }}{E_{{\rm{Gap}}}}  {E_\nu }\) can be interpreted as an effective ionization potential of excitons that corresponds to the exciton binding energy, see also the inset in Fig. 5. It is obvious from Saha’s equation that a large exciton binding energy and low temperature favor the formation of excitons vs. the dissociation into an unbound electronhole plasma. The ionization potential depends on excitation density as a consequence of the excitationinduced lowering of the band continuum edge given by ΔE _{Gap} and the shift of the boundstate energy E _{ ν }. The boundstate shift on the other hand is a net result of bandgap shrinkage, screening of exciton binding energy and Pauli blocking^{33} and is much weaker than the bandgap shift due to compensation effects. In the end, the ionization potential is lowered with increasing excitation density until at \(I_\nu ^{{\rm{eff}}} = 0\) the bound state vanishes and merges with the continuum edge, which is the Mott effect.
A striking observation in Fig. 2 is the degree of ionization approaching unity at low excitation densities, which is somewhat counterintuitive but can be understood from a thermodynamical point of view. The potential that is minimized by the manyparticle system is the free energy F = U − TS. At low densities and fixed temperature, the entropy S gained by a dissociation of an exciton into two separate particles overcompensates the reduction of internal energy U by the exciton binding energy E _{B}. Hence, the socalled entropy ionization already discussed by Mock et al.^{34} is connected to the huge phase space available for quasifree carriers in the lowdensity limit. We may clarify this using the entropy of an ideal gas with N particles in a volume V as given by the SackurTetrode equation:
where c(T) is a temperaturedependent parameter. Obviously, the dissociation of an exciton gas (N particles) into a free electronhole plasma (2N particles) yields the entropy ΔS = Nk _{B}ln(n ^{−1}) with n = N/V up to some additive constant. It follows that the critical density n _{crit} below which the free energy is dominated by entropy essentially scales as exp(−E _{B}/k _{B} T) with temperature.
Excitonic screening
In the spirit of the extended quasiparticle approximation to the spectral function, there are two types of contributions to excitedcarrier screening of the Coulomb interaction, the metallike freecarrier screening and dipolar screening due to bound excitons. The screening can be characterized by the plasmon spectral function, see Eq. (29), that contains excitations in the interacting electronhole plasma as poles in the qωplane, see Fig. 6. In the excitondominated regime shown in Fig. 6a, besides the usual 2d freecarrier contribution at small energies and small momenta, a broad resonance above 150 meV appears. It stems from transitions between 1s and 2s as well as 2plike exciton states, see Fig. 1b, and also from comparable transitions between exciton states with large momenta. There are contributions at smaller energies as well that can not be as easily distinguished from freecarrier screening. At large densities beyond the Mott transition, the plasmon spectral function shows a pronounced peak structure with a squarerootlike behaviour at small momenta, which has been discussed for TMDCs in ref. ^{35} and which is typical for a twodimensional electron gas^{36}. Excitons are expected to be much less polarizable than a free electronhole plasma and, hence, contribute less to screening. Nevertheless, at elevated excitation densities with more than 99% of carriers bound as excitons, their contribution is significant. We demonstrate this by comparing the results for the degree of ionization α _{ a } with and without excitonic screening included using the example of WS_{2} on SiO_{2} substrate. The results are shown in Fig. 7a. Excitonic screening efficiently reduces the ionization potential of excitons at elevated excitation densities, thereby triggering the transition to an ionized plasma, which is reflected by an increase of the degree of ionized carriers. This mechanism of exciton fission is absent when excitonic screening is neglected, thus leading to an ongoing decrease of the ionizedcarrier fraction when coming from the lowdensity side of the ionization curves. From the manyparticle renormalization of the band gap caused by freecarrier and excitonic screening as shown in Fig. 7b, we deduce that in monolayer TMDC semiconductors excitonic screening is less efficient by two orders of magnitude for comparable excitation densities. As the plasmon spectral function is directly observable by electron energy loss spectroscopy^{37}, we suggest to use this technique to explore exciton signatures in the dielectric function eperimentally.
Discussion
Having the composite nature of excitons in mind, photoemission and tunneling spectroscopy on excitons can be seen as semiconductor analogue to induced fission of bosonic particles into their constituents in highenergy collider experiments. Examples include the photodesintegration of deuterium^{38} and the photofission of heavy nuclei^{39}, where nuclear forces instead of Coulomb forces have to be overcome. In timeresolved ARPES, excitons are fissured by photons in the eV range and momentum and energy of electrons as fission products are detected, thereby revealing information about the internal structure of the excitons. In STS, the role of photons as external probe is assumed by a voltage and the spectrum is an average over momentum states. TMDC monolayers offer the unique possibility to optically address different bandstructure valleys selectively^{40}, which we expect to be reflected by the exciton satellites in photoemission studies at short time delays after circularly polarized excitation.
Although the extended quasiparticle approximation and the chemical picture as applied in this paper are very descriptive, we have to be aware of their limitations. Firstly, the approach relies on the assumption of a quasiequilibrium of both types of carriers, excitons and quasifree plasma. A mechanism that yields corrections to this quasiequilibrium picture is the electronhole recombination, either radiative or nonradiative, that reduces the excitation density on a ps time scale^{22}. Given the fact that the relaxation and equilibration of excitons and quasifree carriers are much faster than this^{5,20,41}, empty states are immediately refilled and the system practically remains in quasiequilibrium, where the excitonplasma balance adiabatically adapts to the timedependent density. This picture is also applied in ref. ^{22}, where a ratio between excitons and plasma is assigned to the experimental data at each time during carrier recombination.
A rather fundamental discussion is concerned with the Mott transition as a firstorder phase transition between an exciton gas and a fully ionized electronhole plasma^{18,19}. The phase transition would be connected to an instability of thermodynamic functions that manifests itself in an ambiguity of α _{ a } in a certain region below the Mott density. Due to excitationinduced broadening of the twoparticle states, which is assumed small in our approach, and the shrinkage of the ionization potential towards the Mott transition, quasifree and bound carriers cannot really be separated in this density regime. We avoid this regime as a more sophisticated theory including full spectral functions and excitonexciton interaction would be required. Also, screening in a correlated manyparticle system near the Mott transition is an intricate problem^{42}.
A prominent feature of TMDC semiconductors is the formation of trions, which could in principle be included as additional particle species in the spirit of the chemical picture^{17}. In practice, obtaining boundtrion spectra on the same footing as excitons is a very challenging task on its own, which is beyond the scope of this paper. Qualitatively, one can expect a coexistence of all three phases at room temperature, where some of the exciton population will be drawn to boundtrion states. As trion binding energies are typically in the range of 30 meV^{43}, which is an order of magnitude less than exciton binding energies, this thermal redistribution of population will be much smaller than between excitons and plasma.
Beyond the monolayer limit, TMDC semiconductors can be used to construct bilayer systems^{44} that allow for the formation of spatially indirect, longlived excitons. These systems are expected to be well suited to realize exotic compositeboson phases like condensates at recordhigh temperatures. Our materialrealistic approach to the ionization equilibrium may be extended to include exciton condensates and used to study the complex phase diagrams of bilayer systems. Here, in analogy to electronhole GaAs/InGaAs bilayers, signatures of exciton fission may also be observed in the temperature dependence of the Coulomb drag effect^{45}.
In conclusion, the ionization equilibrium between the fission and fusion of excitons and electronhole pairs in monolayer TMDC semiconductors has been studied for various material as well as experimentally and devicerelevant external parameters on the basis of an ab initio description of the electronic band structure and Coulomb interaction. We observe entropy ionization of excitons at low excitation densities and a Mott transition to a fully ionized plasma at high densities between 3 × 10^{12} cm^{−2} and 1 × 10^{13} cm^{−2}, depending on experimental parameters. Below the Mott transition, excitons become dominant in all cases with maximal fractions of excitons between 70 and >99.9%. The most efficient tuning knobs are dielectric screening of the Coulomb interaction via the choice of dielectric environment and carrier doping that can induce complete ionization above a level of 10^{13} cm^{−2}. Moreover, we find that excitonic screening, although two orders of magnitude less efficient than freecarrier screening at comparable excitation densities, plays an important role in the description of ionization equilibrium. We suggest that fingerprints of excitonic contributions can be observed in ARPES and STS experiments, which are sensitive to the singleparticle spectral functions, thus containing information about the degree of exciton fission and the extent of exciton wave functions in reciprocal space.
Methods
Theory of ionization equilibrium
We start from the general expression for the carrier density (1) and the spectral function
In the limit of small quasiparticle damping \({\rm{Im}}{\kern 1pt} {{\rm{\Sigma }}^{{\rm{ret}},a}} \ll {\rm{Re}}{\kern 1pt} {{\rm{\Sigma }}^{{\rm{ret}},a}}\), the spectral function can be expanded in linear order of Im Σ^{ret,a} yielding the carrier density in socalled extended quasiparticle approximation
where the quasiparticle energy \(E_{{\bf{k}}\sigma }^a\) is given by \(E_{{\bf{k}}\sigma }^a = \varepsilon _{{\bf{k}}\sigma }^a + {\rm{Re}}{\kern 1pt} {\rm{\Sigma }}_{{\bf{k}}\sigma }^{{\rm{ret}},a}(E_{{\bf{k}}\sigma }^a)\) and \({\cal P}\) denotes the Cauchy principal value^{14,19}. The total density is divided into contributions from quasifree particles and correlated particles, the latter being either in bound or scattering manyparticle states.
The spectral function in extended quasiparticle approximation corresponding to this separation into free and correlated carriers is given by
with \({\rm{\Gamma }}_{{\bf{k}}\sigma }^a(\omega ) = {\rm{Im}}{\kern 1pt} {\rm{\Sigma }}_{{\bf{k}}\sigma }^{{\rm{ret}},a}(\omega )\frac{1}{\pi }\frac{{\rm{d}}}{{{\rm{d}}\hbar \omega }}\frac{{\cal P}}{{\hbar \omega  E_{{\bf{k}}\sigma }^a}}\) and the renormalization factor \(Z_{{\bf{k}}\sigma }^a = {\int} {{\rm{d}}\hbar \omega {\rm{\Gamma }}_{{\bf{k}}\sigma }^a(\omega )} \). The first term describes quasifree particles at renormalized energies. Their spectral weight is reduced according to the renormalization factor to account for correlated carriers, which are spectrally described by the second term.
To evaluate the expressions (11) and (12), we have to choose an approximation for the selfenergy Σ^{ret,a}(ω). The real and imaginary parts of Σ determine the quasiparticle energies and the correlated part of the carrier density, respectively. An appropriate choice is the screened ladder approximation^{14,17,46} Σ(ω) = Σ^{H} + Σ^{GW}(ω) + Σ^{T}(ω) that takes into account screening of Coulomb interaction due to excited carriers as well as the formation of bound twoparticle states and consists of Hartree, GW and Tmatrix contributions. We assume that renormalizations due to the Hartree selfenergy are small compared to exchange and correlation effects. In the Tmatrix contribution, we neglect exchange terms and assume static screening so that the Tmatrix depends only on one instead of three frequency arguments. Thus, we obtain for the imaginary part of the selfenergy using the generalized Kadanoff–Baym ansatz^{18}:
Here we applied thermal equilibrium relations for the screened Coulomb interaction^{18}:
\(\varepsilon _{\bf{q}}^{  1,{\rm{ret}}}(\omega )\) is the longitudinal dielectric function describing screening due to excited carriers and n ^{B}(ω) is the Bose distribution function of the elementary plasma excitations called plasmons. V ^{ab} denotes Coulomb matrix elements between species a and b which contain dielectric screening due to carriers in the ground state and due to the environment but no screening due to excited carriers. T′′ denotes the Tmatrix with the two lowestorder terms subtracted from the ladder expansion and is discussed in the following subsection.
Tmatrix and bound carriers
The Tmatrix in statically screened ladder approximation describing bound and scattering twoparticle states between carrier species a and b obeys a Lippmann–Schwinger equation (LSE)
where \({{\cal G}^{{\rm{ret}},ab}}(\omega )\) is the free twoparticle Green’s function in the particleparticle channel. The corresponding interacting twoparticle Green’s function \(G_2^{{\rm{ret}},ab}(\omega )\) fulfills a Bethe–Salpeter equation (BSE), that has been discussed in detail in refs. ^{42,47} and is equivalent to the LSE. We will exploit this fact later when solving the LSE and evaluating the Tmatrix selfenergy. In its homogeneous form, the BSE in static ladder approximation is given by
Diagonalization yields bound states νσσ′Q〉 and eigenenergies \(E_{\nu {\bf{Q}}}^{\sigma \sigma \prime }\). We drop the indices a and b here, assuming that only twoparticle states between different carrier species are involved. Due to the translational invariance of the crystal, the bound states can be classified by the total exciton momentum Q as discussed in^{48}. Here we neglect the effect of electronhole exchange interaction that leads to a finestructure splitting of excitons and trions^{48,49,50} in the meV range, which is small compared to the excitonbinding energies of several hundred meV. As a consequence, electron and hole spins, which are already good quantum numbers in monolayer TMDC materials due to crystal symmetry, also classify the bound states. For each total momentum and spin combination a series of excitons exists, which is labeled here by ν, analogue to the angular momentum states of Hydrogenlike Wannier excitons. Due to the twodimensional nature of monolayer TMDCs and the related strong momentum dependence of dielectric screening, nontrivial exciton series deviating from a hydrogenlike spectrum are observed^{2,51,52}. The eigenenergies approximately decompose into a part from the relative motion of electron and hole and a kinetic part depending on the total momentum: \(E_{\nu {\bf{Q}}}^{\sigma \sigma \prime } = E_{{\rm{rel}},\nu }^{\sigma \sigma \prime } + E_{{\rm{kin}},{\bf{Q}}}^{\sigma \sigma \prime }\). We can use Bloch basis functions to find a representation of the bound states corresponding to exciton wave functions
where k conventionally denotes the hole momentum, while the electron momentum is fixed via the total momentum.
An explicit expression for the Tmatrix can be obtained by writing the LSE (15) in the basis of twoparticle eigenstates νσσ′Q〉 as shown in detail in ref. ^{18}. Since the BSE represents a generalized eigenvalue problem, the eigenstates form a biorthogonal basis. The procedure yields a spectral representation of the Tmatrix in operator form that is referred to as “bilinear expansion”:
with the Pauli blocking factor N _{ ab } = 1 − f ^{a} − f ^{b}, the operator of kinetic energy of unbound electrons and holes E _{kin} and the eigenstate of the adjoint BSE \(\left \langle {\nu \widetilde {\sigma \sigma \prime }{\bf{Q}}} \right\). The bilinear expansion is used in the following to evaluate the imaginary part of the selfenergy (13) and thereby the contribution of correlated carriers.
Separation of bound and quasifree carriers
Inserting Eq. (13) into Eq. (11) and noting that neither the GW selfenergy nor the two lowest Tmatrix terms contribute to the carrier density^{46}, we obtain^{17,19}
\(n_{ab}^B(\omega ) = {[{\rm{exp}}(\beta (\hbar \omega  {\mu _a}  {\mu _b}))  1]^{  1}}\) is the Bose distribution function depending on the chemical potentials of both carrier species. Equation (19) contains contributions of both bound twoparticle states, which are below the singleparticle gap E _{Gap}, and scattering twoparticle states. The latter are explicitly given in refs. ^{15,16,17}. Different excitons are localized at different positions in the Brillouin zone as expressed by the exciton wave functions, Eq. (17), where electrons and holes are separated by the corresponding total momentum Q. Therefore, we do not rely on a global (kindependent) band gap to decide whether a twoparticle state is a bound state. Instead, we compare the energy of each twoparticle state to the sum of electron and hole band energies at the maximum of the respective exciton wave function. The renormalization factor \(Z_{{\bf{k}}\sigma }^a\) of the quasiparticle resonance in the spectral function (12) enters the contribution of correlated carriers as Pauliblocking factor and as correction to the twoparticle scattering spectrum. To simplify the following discussion, we neglect the contribution n _{scatt} of scattering states beyond the quasifree carriers and consider only the boundstate contribution given by the realfrequency poles of the Tmatrix^{15}:
Using Eq. (20), we arrive at the final expression for the carrier density:
The total carrier density separates into contributions from quasifree carriers and from carriers bound as excitons according to the two poles in the spectral function A ^{a}(ω). For a specific material, the ionization equilibrium has to be computed numerically. The electron and hole chemical potentials are determined by adapting the Fermi functions \({f^a}(E_{{\bf{k}}\sigma }^a)\) of electrons and holes to a given density of quasifree carriers at the quasiparticle energies \(E_{{\bf{k}}\sigma }^a\). As the chemical potentials also enter the boundcarrier density via the Bose function \(n_{ab}^B\), Eq. (21) represents an implicit equation for the fraction of quasifree carriers \({\alpha _a} = n_{{\rm{free}}}^a/{n_a}\), that has to be solved selfconsistently with the quasiparticle energies in GW approximation, see Eq. (25), and the boundstate energies \(E_{\nu {\bf{Q}}}^{\sigma \sigma \prime }\). To simplify the procedure, we exploit the fact that shifts of excitonic resonances are naturally much smaller than bandgap shifts, which is due to compensation effects between gap shrinkage and bindingenergy reduction^{33,47}. Hence, we assume that the exciton spectrum depends only weakly on the excitation density so that we can limit ourselves to the BSE (16) in the limit of zero excitation density.
Consistent with the imaginary part of the selfenergy (13), the quasiparticle energies \(E_{{\bf{k}}\sigma }^a\) contain GW and Tmatrix contributions:
The GW selfenergy can be separated into the Fock term and the socalled Montroll–Ward term containing all contributions beyond bare exchange interaction. The Tmatrix contribution is explicitly given in ref. ^{46} and leads to a blue shift of singleparticle energies that is in the nondegenerate case (\({f^a}(E_{{\bf{k}}\sigma }^a) \ll 1\)) caused by the boundcarrier population. At the same time, the Fock selfenergy contains exchange interaction with both quasifree and bound carriers via the extended spectral functions that leads to a lowering of singleparticle energies. This can be seen by using the Tmatrix selfenergy in Eq. (13) to obtain an excitonic contribution to the spectral function (12) given by
It yields a sharp resonance for each bound state weighted by its Bose population function and the exciton wave functions at the corresponding position in kspace. Note that the spectral positions of the resonances are not given by the boundstate energies \(E_{\nu {\bf{Q}}}^{\sigma \sigma \prime }\), which are twoparticle quantities, but by an effective binding energy of the carrier in state k σa〉, as Γ represents a singleparticle spectral function. The Fock selfenergy^{18} can then be expressed in terms of the spectral function using the Kubo–Martin–Schwinger relation for the propagators G ^{<}(ω) in thermal equilibrium:
The first contribution to the Fock selfenergy scales, besides the Coulomb matrix elements, with the freecarrier density, while the second contribution scales with the density of bound carriers. It turns out that similar to exchange interaction with free carriers, boundcarrier exchange leads to kdependent renormalizations according to the exciton wave functions and populations that are contained in the population factor \(f_{{\bf{k}}\prime \sigma }^{a,{\rm{bound}}}\). As a conclusion, the real part of the selfenergy (22) contains quasiparticle renormalizations due to exciton populations via the Tmatrix in two different places that act in opposite directions. We assume that these renormalizations cancel to a large degree and focus on the freecarrier contributions in accordance with refs ^{17,19}. Then we obtain for the quasiparticle energies:
with the Montroll–Ward contribution
The quasiparticle damping \(\gamma _{{\bf{q}},\sigma }^a =  {\rm{Im}}{\kern 1pt} {\rm{\Sigma }}_{{\bf{k}}\sigma }^{{\rm{MW}},{\rm{ret}},a}(E_{{\bf{k}}\sigma }^a)\) is obtained from the selfconsistent evaluation of the GW selfenergy. It is only used for the purpose of calculating the quasiparticle energies, while the spectral function in extended quasiparticle approximation, Eq. (12), involves quasiparticle energies without broadening by construction. We assume that this is valid in a system with continuous density of states. In a similar manner as for the Fock selfenergy, extended spectral functions could be used to evaluate the Montroll–Ward selfenergy in Eq. (26). Due to the spectral structure of the selfenergy, however, renormalizations of the singleparticle band structure caused by bound carriers involve a denominator of the order of the exciton binding energy, which is very offresonant. Therefore the Montroll–Ward term is evaluated using spectral functions for quasifree carriers.
Screening due to excited carriers
In the spirit of the extended quasiparticle approximation, dynamical screening of the Coulomb interaction due to both free carriers and bound excitons is taken into account. The freecarrier screening is treated in RPA with a macroscopic Lindhard dielectric function^{18}, while the excitonic polarizibilities are calculated as described in refs ^{53,54}:
with matrix elements
and exciton wave functions \(\psi _{\nu {\bf{Q}}}^{\sigma \sigma \prime }({\bf{p}})\) as defined above. The momentum and frequency dependence of screening is characterized by the plasmon spectral function
Fraction of bright excitons
In the ionization equilibrium between fusion of unbound carriers and fission of excitons, all bound twoparticle states with quantum number ν and total exciton momentum Q take part, as expressed by the exciton density, see Eq. (21). On the other hand, besides higherorder processes, only excitons with small momenta are optically active, as excitonphoton interaction requires energy and momentum conservation. From the results on the ionization equilibrium discussed in the main text, we extract the fraction of bright excitons by using the obtained electron and hole chemical potentials and summing over the appropriate exciton states in Eq. (21). To numerically resolve the exciton population function in the small window of allowed momenta, we apply an effectivemass approximation to the exciton dispersions shown in Supplementary Fig. 2 and evaluate the sum over exciton states in polar coordinates. The lowest boundstate energies involving electrons and holes with equal spins are thus given by E _{ Q } ≈ E _{1s } + ħ ^{2} Q ^{2}/2M, where we find E _{1s } = −311 meV and M = 1.07m _{e} for MoS_{2} and E _{1s } = −258 meV and M = 0.72m _{e} for WSe_{2}. The energies E _{1s } are measured with respect to the quasiparticle band gap, while energy and momentum conservation in the exciton–photon interaction explicitly involve the band gap, restricting excitons to inside the light cone with radius Q _{max}: ħcQ _{max} = E _{1s } + E _{Gap}. The energy values on the right hand side correspond to the position of the A exciton in optical spectra, which we take from experiment^{55} yielding 1.95 eV and 1.75 eV for MoS_{2} and WSe_{2} on SiO_{2}, respectively. The brightexciton density is given by
taking into account both K and K′ valleys.
Numerical details
We calculate the ionization equilibrium from the fraction of quasifree carriers as root of the implicit Eq. (21). The two highest valence and two lowest conduction bands are considered to cover all excitons that are relevant in a quasiequilibrium situation. Band structures and Coulomb matrix elements are obtained from ab initio calculations as discussed in Supplementary Notes 1 and 2 and illustrated by Supplementary Fig. 1 and Supplementary Table 1. We limit the Brillouin zone to disks with radius 2.7 nm^{−1} around the K, K′, Σ, Σ′ and Γ points using a Monkhorst–Pack mesh with 30 mesh points along ΓM, which yields reasonable convergence of all results. The frequency integrals involved in the Montroll–Ward selfenergy (25) are extended from −600 to 600 meV exploiting the relation \(V_{{\bf{kk}}\prime {\bf{kk}}\prime }^{{\rm{S}}, < ,ab}(  \omega ) = V_{{\bf{kk}}\prime {\bf{kk}}\prime }^{{\rm{S}}, >,ab}(\omega )\). For simplicity, we use a dielectric function (27), which is isotropic in momentum by evaluating its dependence on q along the contour ΓK and using Coulomb matrix elements V _{q} that are averaged over Wannier orbitals, see the Supplementary Information. Both the Lindhard and the excitonic dielectric function (27) are evaluated using groundstate energies and extrapolated to the limit of vanishing phenomenological quasiparticle broadening γ. The excitonic dielectric function is evaluated for momenta q on the Monkhorst–Pack mesh using Eq. (27) and interpolated at arbitrary values of q using cubic Hermite splines. To reach numerical convergence of the dielectric function, we include up to 4000 bound states depending on the physical parameters. The eigenstates and eigenvalues of the BSE, Eq. (16), are obtained by diagonalization using the SLEPc package^{56} for the PETSc toolkit^{57}.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We acknowledge financial support from the Deutsche Forschungsgemeinschaft (JA 141 and RTG 2247 “Quantum Mechanical Materials Modelling”) and the European Graphene Flagship as well as resources for computational time at the HLRN (Hannover/Berlin). M.R. is grateful to the Alexander von Humboldt Foundation for support. We thank Michael Lorke, Christopher Gies, Paul Gartner and Dirk Semkat for fruitful discussions.
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A.S.: Performed the analytical and numerical calculations concerning the ionization equilibrium with support from M.F. M.R. and G.S.: Performed the ab initio calculations. All authors contributed to the interpretation of the results and to the writing of the manuscript.
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Steinhoff, A., Florian, M., Rösner, M. et al. Exciton fission in monolayer transition metal dichalcogenide semiconductors. Nat Commun 8, 1166 (2017). https://doi.org/10.1038/s41467017012986
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