Fast Exciton Annihilation by Capture of Electrons or Holes by Defects via Auger Scattering in Monolayer Metal Dichalcogenides

Fast Exciton Annihilation by Capture of Electrons or Holes by Defects via Auger Scattering in Monolayer Metal Dichalcogenides

Haining Wang, Jared H. Strait, Changjian Zhang, Weimin Chan, Christina Manolatou, Sandip Tiwari, Farhan Rana School of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14853
Abstract

The strong Coulomb interactions and the small exciton radii in two-dimensional metal dichalcogenides can result in very fast capture of electrons and holes of excitons by mid-gap defects from Auger processes. In the Auger processes considered here, an exciton is annihilated at a defect site with the capture of the electron (or the hole) by the defect and the hole (or the electron) is scattered to a high energy. In the case of excitons, the probability of finding an electron and a hole near each other is enhanced many folds compared to the case of free uncorrelated electrons and holes. Consequently, the rate of carrier capture by defects from Auger scattering for excitons in metal dichalcogenides can be 100-1000 times larger than for uncorrelated electrons and holes for carrier densities in the - cm range. We calculate the capture times of electrons and holes by defects and show that the capture times can be in the sub-picosecond to a few picoseconds range. The capture rates exhibit linear as well as quadratic dependence on the exciton density. These fast time scales agree well with the recent experimental observations Shi13 (); Lagarde14 (); Korn11 (); Wangb14 (), and point to the importance of controlling defects in metal dichalcogenides for optoelectronic applications.

I Introduction

Many body interactions play an important role in determining the electronic and optoelectronic properties of two-dimensional (2D) transition metal dichalcogenides (TMDs). The exciton binding energies in 2D chalcogenides are almost an order of magnitude larger compared to other bulk semiconductors Fai10 (); Xu13 (); Changjian14 (); timothy (); Chernikov14 (). The strong Coulomb interactions and small exciton radii in 2D-TMDs result in large optical oscillator strengths Changjian14 (); Konabe14 (); Berg14 () and short radiative lifetimes Wanga14 (). In this paper we show that the same factors also result in very fast capture of electrons and holes of excitons by defects from Auger processes leading to fast non-radiative recombination rates. The basic idea can be understood as follows. Consider the Auger process in which a hole (in the valence band) scatters off an electron (in the conduction band) and is captured by a mid-gap defect level and the electron (in the conduction band) takes the energy released in the hole capture process. In the case of uncorrelated electrons and holes, the rate for this process is proportional to the product of the hole density and the probability of finding an electron near the hole, which is proportional to the electron density . But in the case of tightly bound excitons, an electron is present near the hole with a very high probability proportional to , where is related to the exciton wavefunction (see the discussion below). Therefore, the rate for a hole (or an electron) in a tightly bound exciton to get captured by a defect is proportional to the exciton density times . Generally speaking, Auger rates in semiconductors are considered to be important only at large carrier densities Landsberg92 (). But given the small exciton radii in 2D-TMDs (in the 7-10 range), , which is inversely proportional to the square of the exciton radius, can be extremely large and, consequently, Auger capture rates in 2D-TMDs can be very fast. Compared to the rates for direct electron-hole recombination via interband Auger scattering (exciton-exciton annihilation), which can be limited by the orthogonality of the conduction and valence band Bloch states, the rates for the capture of electrons and holes of excitons by defects can be very fast when the defect states have a good overlap with the conduction or valence band Bloch states.

Quantum efficiencies of TMD light emitters and detectors that have been reported are extremely poor; in the .0001-.01 range Lopez13 (); Ross14 (); Yin12 (); Steiner13 (); Pablo14 (). Similar quantum efficiencies for TMDs have been observed in photoluminescence experiments Fai10 (); Feng10 (); Wangb14 (). Therefore, most of the electrons and holes injected electrically or optically in TMDs recombine non-radiatively. Given that the average radiative lifetimes of excitons in TMDs are in the range of hundreds of picoseconds to a few nanoseconds Wanga14 (), the non-radiative recombination or capture times in TMDs are expected to be of the order of a few picoseconds. Several experimental results on the ultrafast carrier dynamics in photoexcited monolayer MoS do indeed point to non-radiative recombinaton and/or capture times in the few picoseconds range Shi13 (); Korn11 (); Lagarde14 (); Wangb14 (). The mechanisms by which electrons and holes recombine non-radiatively and/or are captured by defects, and the associated time scales, remain to be clarified. The results in this paper show that electrons and holes of excitons in TMDs can get captured by defects on very short times scales that are in the sub-picosecond to a few picoseconds range resulting in exciton annihilation. The capture rates exhibit linear as well as quadratic dependence on the exciton density. The quadratic dependence of the exciton annihilation rate on the exciton density is generally considered to be an exclusive characteristic of exciton-exciton annihilation processes via interband Auger scattering. Although the discussion in this paper focuses on monolayer MoS, the analysis and the results presented here are expected to be relevant to all 2D-TMDs, and are expected to be useful in designing metal dichalcogenide optoelectronic devices as well as in helping to understand and interpret experimental data Shi13 (); Lagarde14 (); Korn11 (); Wangb14 ().

Ii Theoretical Model

ii.1 Introduction

The two basic Auger processes for the capture of an electron (a) or a hole (b) of an exciton by a defect state are depicted in Fig.1. Proper partitioning of the Hamiltonian is important in order to compute the rates of these processes. We discuss the terms in the Hamiltonian describing various processes below.

ii.2 The Non-interacting Hamiltonian

The crystal structure of a monolayer of group-VI dichalcogenides (e.g. =Mo,W and =S,Se) consist of -- layers, and within each layer the atoms (or the atoms) form a 2D hexagonal lattice. Each atoms is surrounded by 6 nearest neighbor atoms in a trigonal prismatic geometry with symmetry. The valence band maxima and conduction band minima occur at the and points in the Brillouin zone. Most of the weight in the conduction and valence band Bloch states near the and points resides on the d-orbitals of atoms Falko13 (); yao12 (); timothy (). The spin up and down valence bands are split near the and points by 0.1-0.2 eV due to the spin-orbit-coupling Falko13 (); yao12 (); timothy (); Liu14 (). In comparison, the spin-orbit-coupling effects in the conduction band are much smaller Liu14 (). Assuming only d-orbitals for the conduction and valence band states, and including spin-orbit coupling, one obtains the following simple spin-dependent tight-binding Hamiltonian (in matrix form) near the () points yao12 (),

 [Δ/2ℏvk−ℏvk+−Δ/2+λτσ] (1)

Here, is related to the material bandgap, stands for the electron spin, stands for the and valleys, is the splitting of the valence band due to spin-orbit coupling, , and the velocity parameter is related to the coupling between the orbitals on neighboring atoms. From density functional theories Lam12 (); Falko13 (), m/s. The wavevectors are measured from the () points. The d-orbital basis used in writing the above Hamiltonian are and  yao12 (). We will use the symbol for the combined valley () and spin () degrees of freedom. Defining as , the energies and eigenvectors of the conduction and valence bands are yao12 (); Efimkin13 (),

 Ecv,s(→k)=λτσ2+γ√(Δs/2)2+(ℏvk)2 (2)
 (3)

Here, (or ) stands for the conduction (or the valence) band, is the phase of the wavevector , and,

 cos(θγ,→k,s)=γΔs2√(Δs/2)2+(ℏvk)2 (4)

Near the conduction band minima and valence band maxima, the band energy dispersion is parabolic with well-defined effective masses, and , for electrons and holes, respectively.

The Hamiltonian describing electron states in the conduction band, valence band, and a mid-gap defect state is,

 Ho = ∑→k,sEc,s(→k)c†→k,sc→k,s+∑→k,sEv,s(→k)b†→k,sb→k,s (5) +∑σEdd†σdσ

Here, , , and are the destruction operators for the conduction band, valence band, and defect states, respectively. The bandgap is . Since only the smallest bandgap will be relevant in the discussion that follows, we will drop the spin/valley indices from for simplicity.

ii.3 Electron-Hole Interaction and Exciton States

The Coulomb interaction between the electrons and holes can be included by adding the following term to the Hamiltonian,

 Heh=1A∑→k,→k′,→q,s,s′V(→q)Fs,s′(→k,→k′,→q)c†→k+→q,sb†→k′−→q,s′b→k′,s′c→k,s (6)

is the 2D Coulomb potential and equals . The wavevector-dependent dielectric constant for monolayer MoS is given by Zhang et al. Changjian14 () and Berkelbach et al. timothy (). is Efimkin13 (),

 Fs,s′(→k,→k′,→q)=⟨vc,→k+→q,s|vc,→k,s⟩⟨vv,→k′−→q,s′|vv,→k′,s′⟩ (7)

Near the conduction band minima, where , and . Similarly, near the valence band maxima, and . Therefore, for wavevectors near the band extrema one can make the approximation Efimkin13 (),

 Fs,s′(→k,→k′,→q)=ei(τϕ→k+→q−τϕ→k+τ′ϕ→k′−τ′ϕ→k′−→q)/2 (8)

Exciton states are approximate eigenstates of the Hamiltonian . Assuming that the ground state of the semiconductor is , which consists of a filled valence band and an empty conduction band, an exciton state with in-plane momentum can be constructed from the ground state as follows Changjian14 (); Efimkin13 (),

 |ψs,s′,α(→Q)⟩=1√A∑→kψα,→Q(→k)c†→k+memex→Q,sb→k−mhmex→Q,s′|ψo⟩ (9)

The exciton wavefunction is . The electron and hole effective masses are and , respectively. The exciton mass is , and the reduced electron-hole mass is . If one writes the exciton wavefunction as,

 ψα,→Q(→k)=ϕα(→k)ei(τϕ→k+(me/mex)→Q+τ′ϕ→k′−(mh/mex)→Q)/2 (10)

then the exciton wavefunction satisfies the standard exciton eigenvalue equation Changjian14 (); timothy (),

 [¯Ec(→k)−¯Ev(→k)]ϕα(→k)−1A∑→qV(→q)ϕα(→k−→q) =Eα(→Q)ϕα(→k) (11)

with an eigenvalue given by,

 Eα(→Q)=Eg−Eα+ℏ2Q22mex (12)

where, is the exciton binding energy. The energy is measured with respect to the energy of the ground state . Note that the phase factors cancel out and do not appear in the exciton eigenvalue equation. The exciton wavefunctions are orthonormal and complete in the sense Kira12 (),

 ∫d2→k(2π)2ϕ∗α(→k)ϕβ(→k)=δα,β (13)
 ∑αϕα(→k)ϕ∗α(→k′)=(2π)2δ2(→k−→k′) (14)

The sum over above includes all the discrete bound exciton states as well as the continuum of ionized exciton states. Finally, the probability of finding an electron and a hole at a distance in the exciton state can be computed by destroying an electron and a hole using the real-space field destruction operators and then taking the overlap of the resulting state with the ground state . The result is where is the Fourier transform of . Note that is not the Fourier transform of , which also includes extra phase factors (see (10).

ii.4 Exciton Basis

In what follows, we will use the exciton basis. The exciton creation operator can be defined as,

 B†s,s′,α(→Q)=1√A∑→kψα,→Q(→k)c†→k+memex→Q,sb→k−mhmex→Q,s′ (15)

Using the completeness and the orthogonality of the exciton wavefunctions given in (14) and (13), we get,

 c†→k,sb→k′,s′=1√A∑αψ∗α,→Q(→kr)B†s,s′,α(→Q) (16)

Here, and equal and on the left hand side, respectively. Products of electron and hole creation and destruction operators can thus be expressed in terms of the exciton operators.

ii.5 Defect States

TMDs (), and in particular Monolayer MoS, are known to have several different kinds of point defects, such as and vacancies and interstitials, impurity atoms, in addition to grain boundaries and dislocations Sofo04 (); Komsa12 (); Seifert13 (); Kong13 (); Noh14 (); Guinea14 (); Robertson13 (); Hao13 (); VanDerZande13 (). The goal in this Section is not to give a detailed description of different defect states in TMDs, something well beyond the scope of this paper, but to capture the essential physics in a way that would enable us to obtain capture rates for electrons and holes and present the main ideas associated with the capture processes.

Since the Bloch states form a complete set, the wavefunction of the electron in the defect state can be expanded in terms of the Bloch states from all the bands Landsberg92 (). In most cases of practical interest, only Bloch states in the vicinity of certain points, , in the Brillouin zone, such as , , and in the case of 2D-TMDs, need to be included in the expansion and therefore one may write,

 ψd(→r)=1√A∑n,→k,scn,s(→k)ei(→Ks+→k).→r√Aun,→k,s(→r) (17)

In the expression above, are the periodic parts of the Bloch functions. The sum over runs over all the energy bands. Whereas shallow defect levels can usually be described well by limiting the summation above to a single band, deep mid-gap defect levels generally have contributions from multiple bands Landsberg92 (); Landsberg80 (). The above expression can usually be cast in much simpler forms for specific defect states.

As an example, we consider the case of the deep point defect in MoS due to a sulfur atom vacancy. A sulfur atom vacancy is a common defect in MoS monolayers and can have a small formation energy Kong13 (); Robertson13 (); Noh14 (). The three states within the bandgap associated with a sulfur vacancy have been obtained previously using ab-initio techniques Kong13 (); Robertson13 (); Noh14 (). These defect states consist of: (i) a single state, made up of mostly the and orbitals of the Mo atoms adjacent to the missing S atom, with an energy few tenths of an eV above the valence band maxima, and (ii) two degenerate states, made up of mostly the , , and orbitals of the Mo atoms adjacent to the missing S atom, with an energy 1.4-1.6 eV above the valence band maxima. All the defect states are spin-degenerate and correspond to the one () and two dimensional () representations of the trigonal symmetry group . The computed orbitals of these states are shown in Fig.2 (from Noh et al. Noh14 ()). A defect state can be an efficient center for non-radiative recombination due to Auger scattering only if it has good overlaps with the Bloch states of both the conduction and the valence bands. The states fit this criterion. The states can be described well by limiting the summation in the expression above to the Bloch states of the conduction and the valence band extrema at the and points. Since all the orbitals forming the states have weights almost entirely on the Mo atoms adjacent to the missing S atom, one may write . Since does not vary much with near the band extrema, the sum in (17) can be rearranged to give,

 ψd(→r)=χd(→r)∑n=c,vsbn,sei→Ks.→reiγτϕ→k––/2un,→k––,s(→r) (18)

Here, the line under means that any wavevector near the band extrema can be chosen. The function is expected to be localized at the defect, becoming very small at the second nearest Mo atom near the defect site.

ii.6 Hamiltonian for the Capture of Holes and Electrons

Consider process (b) in Fig.1 in which a hole scatters off an electron and is captured by a defect and the electron is scattered to a higher energy. The relevant term in the Coulomb interaction Hamiltonian that describes the hole capture process in Fig.1(b) can be written as,

 Hhc=1A∑→k,→k′,→q,s,s′V(→q)Ms,s′(→k,→k′,→q)c†→k+→q,sb†→k′,s′dσ′c→k,s+h.c. (19)

The overlap factor equals,

 Ms,s′(→k,→k′,→q)=⟨vc,→k+→q,s|vc,→k,s⟩ ×∑n=c,vbn,s′eiγτ′ϕ→k––/2⟨vv,→k′,s′|vn,→k––,s′⟩ ×1√A∫d2→rχd(→r)e−i(→k′+→q).→r ≈ei(τϕ→k+→q−τϕ→k−τ′ϕ→k′)/2bv,s′√Aχd(→k′+→q) (20)

Similarly, the electron capture process (Fig.1(a)) is described by the Hamiltonian,

 Hec=1A∑→k,→k′,→q,s,s′V(→q)Ls,s′(→k,→k′,→q)d†σb†→k′,s′b→k′+→q,s′c→k,s+h.c. (21)

where overlap factor equals,

 Ls,s′(→k,→k′,→q)≈ei(τ′ϕ→k′+→q−τ′ϕ→k′−τϕ→k)/2b∗c,s√Aχ∗d(→k+→q) (22)

The potential of the defect does not appear in the Hamiltonian above. The reason for this is that it has already been taken into account in defining the non-interacting Hamiltonian, and its eigenstates, in Section (II.2).

Iii Electron and Hole Capture Rates for Excitons

We assume an initial state described by the density operator in which the exciton occupation , defect occupation , and conduction and valence band occupations are given by,

 ⟨d†σdσ′⟩=fdδσ,σ′ ⟨c†→k,sc→k′,s′⟩=fc,s(→k)δs,s′δ→k,→k′ ⟨b†→k,sb→k′,s′⟩=fv,s(→k)δs,s′δ→k,→k′ ⟨B†s,s′,α(→Q)Bs,s′,α(→Q)⟩=ns,s′,α(→Q)+ 1A∑→k|ϕα(→k)|2fc,s(→k+memex→Q)[1−fv,s′(→k−mhmex→Q)]

The angled brackets stand for ensemble averaging with respect to the the density operator . Since the excitons are not exact bosons, the value of is not just equal to the exciton occupation . Using the cluster expansion to evaluate results in the additional Hartree-Fock term shown above Kira06 (); Koch06 (). The same extra term also shows up in the luminescence spectra of excitons Kira12 (), and, as discussed below, this term results in a quadratic dependence of the capture rate on the exciton density at large exciton densities.

We assume that the electron and hole densities for different spins/valleys (including both free carriers and bound excitons) are and , respectively, and the defect density is . The initial ensemble consists of states that are approximate eigenstates of but not of . Therefore, we consider and as perturbations.

iii.1 Electron Capture Rate

We first consider process (a) in Fig.1 in which the electron is captured by a defect. The average electron capture rate (units: per unit area per second) can be calculated from the first order perturbation theory using the exciton basis described in Section II.4 and the average values given in (III). The details of the calculations are given in the Appendix. The final result is,

 Rec≈2πℏnd(1−fd)∑s,s′,αDv,s′(qα)|χd(qα)|2|bc,s|2 ×∣∣ ∣∣1A∑→krV(qα^x−→kr)ϕα(→kr)∣∣ ∣∣2[ns,s′,α +1A2∑→k,→Q|ϕα(→k)|2fc,s(→k+memex→Q)[1−fv,s′(→k−mhmex→Q)]⎤⎥⎦

Here, is the valence band density of states (per valley per spin) evaluated at the energy of the scattered hole whose wavevector is . is approximately given by the relation, . Note that none of the phase factors appear in the above result. The exciton density is,

 ns,s′,α=∫d2→Q(2π)2ns,s′,α(→Q) (25)

If for all values of for which is significant, then the above expression reduces to,

 Rec = 2πℏnd(1−fd)∑s,s′,αDv,s′(qα)|V(qα)|2|χd(qα)|2 (26) ×|bc,s|2[|ϕα(→r=0)|2ns,s′,α+Gαnsps′]

Expression for is given in the Appendix. is significant for only the lowest few exciton states.

iii.2 Hole Capture Rate

The rate for process (b) in Fig.1 in which the hole is captured by a defect can be calculated in the same way. The result is,

 Rhc≈2πℏndfd∑s,s′,αDc,s(qα)|χd(qα)|2|bv,s′|2 ×∣∣ ∣∣1A∑→krV(qα^x−→kr)ϕα(→kr)∣∣ ∣∣2[ns,s′,α +1A2∑→k,→Q|ϕα(→k)|2fc,s(→k+memex→Q)[1−fv,s′(→k−mhmex→Q)]⎤⎥⎦

where now is approximately given by the relation, . And, as before, if for all values of for which is significant, then the above expression reduces to,

 Rhc = 2πℏndfd∑s,s′,αDc,s(qα)|V(qα)|2|χd(qα)|2 (28) ×|bv,s′|2[|ϕα(→r=0)|2ns,s′,α+Gαnsps′]

iii.3 Coulomb Correlations and Enhancement of the Auger Capture Rates

Equation (26) for the electron capture rate can also be written as,

 Rec = 2πℏnd(1−fd)∑s,s′,αDv,s′(qα)|V(qα)|2|χd(qα)|2 (29) ×|bc,s|2nsps′[Gα+gs,s′,α(→r=0)]

where, . The quantity inside the square brackets in (29), , describes the enhancement in the probability of finding an electron and a hole close to each other as a result of the attractive Coulomb interactions. It is interesting to compare the electron capture rate in (29) with the result obtained assuming no electron-hole attractive interaction (i.e. ),

 Rec = 2πℏnd(1−fd) (30) × ∑s,s′Dv,s′(qo)|V(qo)|2|χd(qo)|2|bc,s|2nsps′

where is approximately given by the relation, . It can be seen that the capture rate in (29) is larger by the same enhancement factor. Assuming all the electrons and holes are in the lowest () bound exciton state, values of and are independent of the valley/spin indices, and the exciton density is , the comparison between (29) and (30) shows that the enhancement of the electron capture rate in the case of excitons is roughly proportional to . Given that the radius of the lowest exciton state in monolayer MoS is in the 7-10 range Changjian14 (), the enhancement, assuming an exciton density of cm, is in the 72-138 range, and in the 644-1308 range if the exciton density is assumed to be cm. Therefore, the correlations in the positions of the electrons and the holes as a result of the attractive Coulomb interaction make electrons and holes in tightly bound excitons in TMDs far more susceptible to capture by defects compared to uncorrelated free carriers. Interestingly, even when the exciton density is zero the capture rate in (29) is enhanced by the factors compared to the rate in (30) for uncorrelated electrons and holes. Therefore, Coulomb correlations in the positions of electrons and holes due to the attractive interaction between them enhances the Auger scattering rates even at the Hartree-Fock level.

Iv Numerical Results and Discussion

iv.1 Carrier Capture Times at Low Exciton Densities

For numerical computations, we consider monolayer MoS on a quartz substrate, as is the case in many experiments. We first assume that the exciton density is small enough ( cm) to allow one to ignore phase-space filling effects Changjian14 (). We use the wavevector-dependent dielectric constant for monolayer MoS on quartz given by Zhang et al. Changjian14 (). The defect state wavefunction is given in (18). The values of and are assumed to be independent of the valley/spin indices. This is a good approximation for many important cases. For example, in the case of the sulfur vacancy in MoS discussed earlier, the states have a total weight of 0.25 on the orbitals of the Mo atoms adjacent to the missing sulfur atom Yong14 (). Since the conduction band Bloch states of both and valleys are made up of mostly the orbitals of Mo atoms, is the same for both the valleys. We approximate the envelope, , of the defect state wavefunction in (18) by a Gaussian, , where (see Fig.2). Note that the in-plane S-Mo bound length in MoS is 1.83 . Fig.3 plots the computed capture times of electrons () and holes () of excitons assuming that all the excitons are in the lowest state (). In the low exciton density limit considered here these capture times are independent of the exciton density. The defect density is assumed to be cm. The capture times for electrons and holes shown in Fig.3 have been normalized by multiplying them by and , respectively, given the uncertainty in the exact values of these parameters. In the calculation of the electron capture times the defect state is assumed to be empty (), and in the calculation of the hole capture times the defect state is assumed to be full ().

The curves shown in Fig.3 can provide results in different situations. For example, in the case of the states associated with a sulfur vacancy, if is assumed to be 0.25 Yong14 (), then the electron capture time curve in Fig.3 would need to be multiplied by 4 in order to get the actual electron capture times. If the state energy is assumed to 1.5 eV above the valence band edge Noh14 (), then the electron capture time comes out to be 2.4 ps. Since the capture times decrease inversely with the defect density , the capture times shown in Fig.3 can be interpolated for different values of the defect density. For example, a defect density of cm would result in an electron capture time of 0.6 ps for the state of a sulfur vacancy (under the same assumptions as stated above).

Fig.3 shows that shallower traps have much shorter capture times than deeper traps. This can be understood as follows. Energy conservation requires that the scattered electron (hole), in a hole (electron) capture process, takes away most of the energy. The deeper the trap the more the final energy of the scattered particle. Also, momentum conservation requires that the momentum of the scattered particle be provided by the relevant Fourier component of the defect state wavefunction. Therefore, the deeper the trap the larger the momentum transfer. Since in Fourier space the defect state wavefunction is , larger momentum transfers result in smaller capture rates. Note that this result is largely independent of the exact assumed form of the defect state wavefunction. In addition, the Coulomb potential also decreases for larger momentum transfers. Although the final density of states available to the scattered particle increases with the particle energy (for non-parabolic energy band dispersions in 2D), this increase is not enough to offset the reduction in the capture rates due to the factors mentioned above.

Since the energy width of the valence and conduction bands in MoS are less than 1.2 eV and 0.6 eV Lam12 (); Louie1 (), respectively, the limited horizontal extents of the curves in Fig.3 ensure that the electron (hole) scattered to a high energy in the hole (electron) capture process is scattered within the same band consistent with the assumptions made in this work. It is, however, possible for the scattered particle to go into a different band. For example, slightly away from the () points, the next higher conduction band has Bloch states with a large weight on the orbitals of Mo atoms and these Bloch states will have large overlap with the Bloch states near the conduction band bottom Guinea13 (). It should also be noted that the weights and for defects could be very small or zero. For example, in the case of sulfur vacancy states both and are expected to be very small Robertson13 (); Noh14 (); Yong14 ().

iv.2 Carrier Capture Times at High Exciton Densities

At large exciton densities (typically larger than cm, but smaller than cm, for 2D-TMDs Changjian14 ()), phase-space filling effects cannot be ignored in the description of the exciton states. We use the formalism developed by Kira and Koch Kira12 (); Kira06 (). When phase-space filling is taken into account, exciton eigenvalue equation in the relative co-ordinates becomes non-Hermitian (see the Appendix) and its solutions are expressed in terms of the left and the right eigenfunctions, and , respectively. These eigenfunctions are a also a function of the center of mass momentum , and are related as follows Kira12 (); Kira06 (),

 ϕRs,s′,α(→k,→Q)=ϕLs,s′,α(→k,→Q)[fv,s′(→k−mhmex→Q) −fc,s(→k+memex→Q)] (31)

and obey the orthogonality relation,

 ∫d2→k(2π)2[ϕLs,s′,α(→k,→Q)]∗ϕRs,s′,β(→k,→Q)=δα,β (32)

In terms of these eigenfunctions, the expression for the electron capture rate becomes,

 Rec≈2πℏnd(1−fd)1A∑s,s′,α,→QDv,s′(qα)|χd(qα)|2|bc,s|2 ×∣∣ ∣∣1A∑→krV(qα^x−→kr)ϕRs,s′,α(→kr,→Q)∣∣ ∣∣2[ns,s′,α(→Q) +1A∑→k|ϕLs,s′,α(→k,→Q)|2fc,s(→k+memex→Q) ×[1−fv,s′(→k−mhmex→Q)]]

The expression for the capture rate of holes in the high exciton density case follows similarly from (LABEL:eq:Rhc_0). When all electrons and holes exist as excitons, self-consistency requires that the distribution functions are given by Kira12 (),

 fc,s(→k)=1A∑s′,α,→Q[ϕLs,s′,α(→k,→Q)]∗ϕRs,s′,α(→k,→Q)ns,s′,α(→Q) 1−fv,s′(→k)=1A∑s,α,→Q[ϕLs,s′,α(→k,→Q)]∗ϕRs,s′,α(→k,→Q)ns,s′,α(→Q)

Equations (IV.2) and (IV.2) show that the capture rate has terms that go linearly as well as quadratically with the exciton density. The quadratic dependence comes from the Hartree-Fock term in the evaluation of (see Equation (III)). It can be understood as coming from the Auger scattering between the electron of one exciton and the hole of another exciton. Recall from the discussion in Section III.3 that even at the Hartree-Fock level Auger scattering between electrons and holes is enhanced due to the Coulomb correlations compared to uncorrelated electrons and holes.

For numerical computations, we again consider monolayer MoS on a quartz substrate, as in Section IV.1. We solve the exciton eigenvalue equation for different exciton densities and obtain the exciton radii and the exciton binding energies Changjian14 (). For simplicity, we consider the case when all the electrons and holes are in the lowest () bound exciton state. Fig.4 plots the inverse capture time () of the electron of an exciton in monolayer MoS on quartz as a function of the exciton density. The plotted capture time has been normalized by multiplying it by