A fully microscopic model of the doping-dependent exciton and trion linewidths in the absorption spectra of monolayer transition metal dichalcogenides in the low temperature and low-doping regime is explored. The approach is based on perturbation theory and avoids the use of phenomenological parameters. In the low-doping regime, we find that the trion linewidth is relatively insensitive to doping levels, while the exciton linewidth increases monotonically with doping. On the other hand, we argue that the trion linewidth shows a somewhat stronger temperature dependence. The magnitudes of the linewidths are likely to be masked by phonon scattering for T ≥ 20 K in encapsulated samples in the low-doping regime. We discuss the breakdown of perturbation theory, which should occur at relatively low-doping levels and low temperatures. Our work also paves the way toward understanding a variety of related scattering processes, including impact ionization and Auger scattering in clean 2D samples.

Monolayer transition metal dichalcogenides (TMDCs) are quasi-two-dimensional materials known to exhibit extraordinary physical phenomena.1,2 These materials may be viewed as semiconducting analogs of graphene3–5 and present with non-trivial optical, electronic, and, under some circumstances, topological and superconducting properties.6,7 Due to their unique characteristics, monolayer TMDCs have been proposed for myriad practical applications8 such as opto-electronics,9–12 field-effect transistors,13 and digital logic gates.14,15 Of particular fundamental interest is the nature of electron–hole complexes such as excitons16,17 and trions18,19 in TMDCs. Due to the reduced screening in two-dimensional (2D) systems, such stable carrier complexes may have anomalously large binding energies, with that of the exciton reaching ∼0.5 eV16,20–22 and that of the trion reported to be in the range of 20–35 meV,18,21,23–25 implying that trions are bound even at room temperature. These observations indicate that monolayer TMDCs are unique systems for investigating the properties of strongly interacting quasiparticles. In addition, they may provide unprecedented experimental clarity concerning the nature of interactions between these electron–hole complexes and phonons26,27 as well as with charge carriers and other quasiparticles.

A standard means of probing the nature of the interactions of excitons and trions with other excitations is via the broadening of linewidths in clean samples with respect to control parameters such as the temperature or carrier density. Intrinsic homogeneous quasiparticle (QP) linewidths26 are generally obfuscated by inhomogeneous broadening due to the high level of static defects in processing. However, recent work has led to the observation of very narrow QP linewidths via the preparation of ultra-clean samples by both dry transfer methods and chemical vapor deposition28–30 and by the usage of non-linear spectroscopy to extract the homogeneous linewidth from inhomogeneously broadened spectra.26 The optical interrogation of the exciton and trion linewidths in these less defective samples offers a unique opportunity to understand the mechanisms of the 2D exciton and trion scattering processes in quasi-2D systems.

There are many factors that affect line broadening in monolayer TMDCs, most notably interactions with phonons (as controlled by temperature) and interactions with other charge carriers (as controlled by doping). At very low temperatures and near the charge-neutrality point,27 it is expected that the intrinsic homogeneous linewidth due to lifetime broadening may be observed if the sample is clean enough. As temperature increases, phonons begin to play a significant role and will eventually dominate the line broadening process. The interaction of excitons with phonons has been studied in some detail in TMDCs,27,31 and a variety of coupling motifs have been elucidated experimentally and theoretically.

Additionally, the concentration of electrons as controlled by gating can alter linewidths and line shapes in a non-trivial fashion.16,32–34 Studies that have investigated the electron density dependence of optical line shapes in monolayer TMDCs from the Fermi-polaron perspective provide a means of describing optical line broadening as a function of doping.35–41 Such many-body multiple scattering theories are essential for properly describing the full range of doping-dependent behavior, as the Fermi golden rule breaks down at sizable doping levels. However, the use of graphene gating and clean samples renders the investigation of the doping regime close to the charge-neutrality point possible.42 Here, detailed microscopic golden rule-based calculations that can provide new insights into the line broadening mechanisms may be performed. Motivated by the aforementioned recent experimental works, we follow this latter path to assess how the elastic scattering of excitons and trions with free charge carriers may alter linewidths of both ground and excited state excitonic complexes in the low-doping regime. In particular, we investigate the circumstances for which doping-related broadening may compete with phonon-induced broadening, and we discuss the breakdown of the perturbative approach as a function of temperature and carrier density. The importance of our work extends beyond the description of linewidths and is of relevance for describing scattering processes such as Auger recombination and impact ionization in TMDCs.

Our paper is organized as follows: We first present an outline of the microscopic theory in Sec. II, focusing on the electron–exciton scattering calculation, which is discussed in Subsection II A. Calculations for the electron–trion scattering are similar to that of the exciton and discussed (briefly) in Subsection II B. The low-temperature results for the trion and exciton linewidths, in addition to the details of the model and limitations of the golden rule approach, are presented and discussed in Sec. III. Finally, in Sec. IV, we summarize our conclusions and discuss outlook and potential future work. Details not contained in the main text are located in several appendixes.

In this section, the elastic (energy-conserving) scattering of electrons from both excitons and trions are described within the Fermi golden rule approximation. Additionally, because we work at the golden rule level of theory, bound states in scattering are not considered. While such a treatment can only be valid at extremely low doping densities, recent synthetic work using encapsulated samples points to a route to experimentally controlled access to this regime. Furthermore, the use of the golden rule allows for a very detailed microscopic description,43,44 the limitations of which will be discussed in the following sections.

In order to facilitate the computation, we use a simple variational guess for the exciton wave function

ϕ(r)=π2λ2er/λ,
(1)

where r is the relative coordinate of the two-body system. The optimal effective Bohr radius λ is chosen to best match the functional form of (1) to the ground state of a Wannier exciton in a Rytova–Keldysh potential45,46 found using exact diagonalization.

The second-quantized form of the exciton-free electron scattering state is

kx,ke=kϕαxkx+k*ψke*ckdkx+kcke0,
(2)

which is a direct product state of the free exciton and electron states, kxke. The wave function

ϕk=8πλ2Ag(λk)
(3)

satisfies normalization kϕk2A(2π)2d2kϕk2=1 and is derived by performing an in-plane Fourier transform of (1), where g(x)=[1+x2]3/2, ck (dk) are electron (hole) annihilation operators for momentum index k, A is the in-plane area of the 2D material, and αx = me/Mx is the ratio of the electron and exciton effective masses (which manifests during the coordinate transform to relative/center of mass coordinates). The free-electron wave function ψkeik·R characterizes an electron that may exhibit free in-plane motion and together with the center of mass coordinate of the exciton, contributes only a global phase factor that may be ignored in subsequent calculations, as it does not contribute to the determination of the scattering rate.

Scattering matrix elements are computed by evaluating the coupling between an initial QP-free electron state, kx,ke, and a final QP-free electron state in which momentum q is transferred, kx+q,keq. The second-quantized, momentum-conserving potential energy operator V = Veh + Vee that mediates this coupling may be split into electron–hole and electron–electron components,

Veh=k1,k2,qs=,vqck1+qsdk2qdk2ck1s
(4a)

and

Vee=12k1,k2,qs1,s2=,vqck1+qs1ck2qs2ck2s2ck1s1,
(4b)

where vq=2πe2Aqε(q) is the magnitude of the two-body interactions and ε(q) is a static dielectric function discussed in Sec. II C. The exciton-free electron elastic scattering matrix elements are henceforth defined as

V(q,ke,kx)=kx+q,keq|V|kx,ke.
(5)

Once matrix elements have been computed, the linewidth Γ(n; kx) is calculated by summing over all final exciton states,

Γ(n,kx)=A(2π)2d2qw(q;n,kx).
(6)

Here, w(q; n, kx) is a partial scattering rate computed for fixed momentum transfer using Fermi’s golden rule,

w(q;n,kx)=2πke|V(q,ke,kx)|2f(ke)[1f(|keq|)]×δ2kx22Mx+2ke22me2|kx+q|22Mx2|keq|22me,
(7)

where the Fermi–Dirac distribution

f(k)=e(2k2/2meμ)/kBT+11
(8)

contains doping-density (n) dependence through the chemical potential μ=kBTlnexpεF/kBT1 and the Fermi energy of a 2D electron gas, εF = πℏ2n/me. In order to simplify the calculations, the parameter kx = 0 is taken in all computations, effectively choosing a reference frame in which the exciton is at rest. For further details, we refer the reader to Ref. 43, where similar calculations are performed for anisotropic 3D systems.

Computation of the trion-free electron elastic scattering linewidth contribution is similar to that of the excitonic case in all ways except for the determination of the scattering matrix elements. The trion-free electron scattering state is constructed similarly to that of (2), with a few key distinctions to be noted below. Explicitly, we write this scattering state as

kt,ke=k1,k2s1,s2,seξS*(s1,s2)Φαtkt+k1,αtkt+k2*ψke*ck1s1ck2s2dkt+k1+k2ckese0.
(9)

Note the introduction of a spin wave function that constrains the trion to the singlet spin configuration, ξS(s1,s2)=s1s2S, via the projection of a two-fermion spin state s1s2 on the singlet state S. The projection satisfies the properties s1,s2ξS*(s1,s2)ξS(s1,s2)=SS=1, and ξS(s1, s2) = −ξS(s2, s1) as per Fermionic anti-commutation rules. Given that the trion triplet state is, at most, weakly bound, we consider only singlet to singlet scattering.

The trion wave function, Φ, is given by

Φk1,k2=N8πλ1λ2Ag(λ1k1)g(λ2k2).
(10)

Here, λ1 and λ2 are variational parameters associated with the Chandrasekhar-type wave function,18,47 and the constant N is a normalization factor,

N=11+κ2,κ=4λ1λ2(λ1+λ2)2,
(11)

that arises during the variational minimization of the trion binding energy.48 

Once the matrix elements

V(q,ke,kt)=kt+q,keq|V|kt,ke
(12)

are computed, the trion linewidth may be determined using (6) and (7) in the same way as for the exciton case (with the appropriate substitutions, e.g., the initial QP momentum kxkt, the mass ratio αxαt = me/Mt, etc.). Linewidths for low doping densities are reported in Sec. III, computational details of this calculation are given in  Appendix D, and the physical parameters used may be found in the caption of Fig. 1.

FIG. 1.

Linewidth broadening of monolayer MoSe2 as a function of electron doping density for BN-encapsulated (ε0 = 4.5)61 monolayers. The following parameters were used: in the exciton calculation, the effective Bohr radii λ0 = 10.3 and in the case of the trion, λ1 = λ0 and λ2 = 25.2 Å.18 In the exciton 2s elastic scattering, a = 7.79 Å and b = 6.20 Å (see Appendix B 4). The electron (hole) effective masses employed were 0.49 (0.61) (in units of m0),62 and the polarizability was χ2D = 8.23 Å.18 In the case of exciton elastic scattering, the singlet and triplet contributions are identical as the exchange contribution to the potential dominates; trion triplet states are not considered. Additionally, screening using the effective frequency-dependent dielectric function [see Eq. (16)] are presented for the trion, as effective screening does not appear to affect the exciton linewidth.

FIG. 1.

Linewidth broadening of monolayer MoSe2 as a function of electron doping density for BN-encapsulated (ε0 = 4.5)61 monolayers. The following parameters were used: in the exciton calculation, the effective Bohr radii λ0 = 10.3 and in the case of the trion, λ1 = λ0 and λ2 = 25.2 Å.18 In the exciton 2s elastic scattering, a = 7.79 Å and b = 6.20 Å (see Appendix B 4). The electron (hole) effective masses employed were 0.49 (0.61) (in units of m0),62 and the polarizability was χ2D = 8.23 Å.18 In the case of exciton elastic scattering, the singlet and triplet contributions are identical as the exchange contribution to the potential dominates; trion triplet states are not considered. Additionally, screening using the effective frequency-dependent dielectric function [see Eq. (16)] are presented for the trion, as effective screening does not appear to affect the exciton linewidth.

Close modal

The dielectric function ε(q, ω) takes into account properties of the monolayer TMDC, the surrounding medium, and the excess electron gas,49 respectively, and may be broken down into distinct contributions as50 

ε(q,ω)=εI(q)+εII(q,ω).
(13)

We follow previous work51 and screen the direct and exchange interactions, in contrast to the usual Bethe-Salpeter treatment of bound state formation where the exchange interaction is unscreened.52–55 The first term consists of a static contribution from the monolayer TMDC and surrounding layers in the absence of doping,

εI(q)=ε0(1+2πχ2Dq),
(14)

where ε0 = (εa + εb)/2 is the dielectric constant of the surrounding medium45,46 (the average of the two encapsulating dielectrics) and χ2D is the dielectric polarizability of the 2D material.

The second term is due to the presence of doping electrons and is generally frequency dependent. We follow Stern49 and treat the excess electrons as a 2D homogeneous electron gas (HEG). In the static (ω = 0) approximation, this yields

εII(q,0)=2me22q1ifq2kF11(2kF/q)2ifq>2kF.
(15)

Note that (15) implicitly carries a doping density (n) dependence through the Fermi momentum pF=kF=2πn.

To motivate this choice, we observe that ε(q, 0) captures the correct behavior in both the small-wavelength and low-doping limits. In the low-doping limit, the Stern-like term vanishes and the dielectric function ε(q, 0) → εI(q), which is the dielectric function of the material and its surroundings. The low q-limit suppresses the term containing the polarizability and diverges like 1/q, correctly screening the 2D Coulomb interaction at small q.56 

If doping levels are large enough, the static approximation presented above will fail.56,57 Although this signals one aspect of the high doping density breakdown of the golden rule, one way to potentially extend its domain of validity is to utilize a frequency-dependent scattering matrix element as discussed in Ref. 51. This leads to a dielectric function derived from the 2D Lindhard function, εII(q, Eeff/),49,51 evaluated at the effective energy

EeffE(ke)E(|keq|)=2(2keqq2)2m0,
(16)

which is the energy difference between the initial and final states of the scattering electron. The details of εII(q, ω) are presented in Ref. 49 and in  Appendix A.

The Fermi golden rule is expected to be valid only in the ultra-low doping regime (εFεt ∼ 1012 cm−2), where εt denotes the trion binding energy in the limit of zero doping. As the doping level increases, many-body, multi-scattering effects become prominent,58 and a Fermi-polaron-like picture appears to be required.35,37,38 Since the low-doping regime is now potentially controllable and accessible in encapsulated samples with graphene gating layers, a golden rule approach is useful in enabling a fully microscopic treatment in this restricted regime.

Linewidths vs doping level for both the exciton and trion lines are displayed for 5 K and 25 K in Fig. 1. Results are presented for the experimentally relevant case of a layer encapsulated by dielectric media with properties mimicking those of boron nitride. We also note that in a hypothetical suspended sample (ε0 = 1), Γ is enhanced compared to results presented in Fig. 1 (e.g., roughly 5 meV at 1011 cm−2, compared to only 1 meV in the encapsulated case) and is comparable to, or even larger than, that associated with phonon-induced broadening, since the scaling of Γ with respect to the background dielectric function varies roughly as ε02. It should also be noted that in experiments the encapsulating layers are of finite thickness, and while this situation can be handled theoretically,59,60 we do not do so here as it complicates the treatment of the dielectric function. We thus expect the true magnitude of linewidth values to be somewhat larger than the values presented in Fig. 1. Additionally, while we have also carried out an investigation of inelastic electron-capture scattering, we find that elastic scattering dominates the linewidths in the regimes we consider. Thus, we only focus on the elastic scattering contributions.

We first discuss trion line broadening. For doping levels n > 0.4 × 1011 cm−2, the trion linewidth in all cases is largely independent of doping density. The upturn seen in the static screening trion linewidth as doping density decreases is likely an artifact of behavior embedded in the function ε2(q). Indeed, a suppression of the q−3 behavior for large q of this function leads to an essentially flat trion linewidth as a function of doping level, similar to that seen in Fermi-polaron-like theories and in some experiments.63,64 It should be noted that in these approaches, however, the trion line broadening is controlled by a phenomenological input parameter. For example, the width of the doping-independent trion line in the work of Efimkin and MacDonald38 is given by the parameter γ, which is input phenomenologically by hand, is not derived directly from the microscopic interactionsin the system.

Here, our fully microscopic approach allows for the microscopic extraction of the magnitude and temperature dependence of the trion linewidth. While the static and effective frequency-dependent screening cases are largely in agreement at low T, the same cannot be said for results at 25 K. Given the subtle changes in the scattering matrix elements except at small q, this difference likely arises from the larger accessible density of states available at higher densities away from ω = 0 in the screening function.

We now turn to the broadening of the exciton line. Unlike the trion case, the exciton linewidth monotonically increases as a function of doping density at low values of n in the 25 K case. This is again in agreement with experimental expectations65–67 as well as the behavior found in many-body approaches.35,38,58 In particular, in these latter theoretical approaches, an approximately linear dependence of the linewidth on doping manifests over a much wider doping density range for the exciton line. The very same behavior arises from the golden rule at extremely low doping. The decrease of the slope of the linewidth as n increases, most clearly demonstrated in the near-plateau of the 5 K exciton linewidths above n = 0.8 × 1011 cm−2, is a signature of the breakdown of the golden rule. Specifically, due to the εF/kBT term in the Fermi–Dirac distribution function, the crossover from the non-degenerate to the degenerate electron gas limit will induce a change in the doping dependence of the excitonic linewidth from a linear scaling Γ ∼ n at low doping to an eventual plateau ∼kBT, and then an unphysical decline with increasing n. This same trend is reported in Ref. 43 for the quantum well case. We systematically examine this behavior in Fig. 2, which shows the doping and temperature dependence of this behavior. If one focuses on the more physically described regime n < 0.8 × 1011 cm−2, it is observed that, unlike in the trion case, doping-induced exciton line broadening is largely insensitive to temperature variations in the range T = 5–25 K. Furthermore, given the fact that phonon-induced line broadening is suppressed at these temperatures, doping-induced line broadening effects may be observable at T = 5 K in clean, encapsulated samples even for doping densities as low as n ∼ 2 × 1011 cm−2, especially with respect to the 2s line, where the line broadening effects appear to be slightly enhanced compared to the ground state.

FIG. 2.

Doping dependence of the 1s exciton linewidth at temperatures T = 5, 6, …, 15 K (left). Parameters describe monolayer MoSe2, as seen in Fig. 1. Lower linewidths correspond to lower temperatures. The horizontal dashed lines show the plateau location. The value of the maximum (plateau) is also plotted as a function of temperature, showing a clearly linear behavior at low T (right).

FIG. 2.

Doping dependence of the 1s exciton linewidth at temperatures T = 5, 6, …, 15 K (left). Parameters describe monolayer MoSe2, as seen in Fig. 1. Lower linewidths correspond to lower temperatures. The horizontal dashed lines show the plateau location. The value of the maximum (plateau) is also plotted as a function of temperature, showing a clearly linear behavior at low T (right).

Close modal

In this work, we have employed perturbation theory to calculate the rates of electron–exciton and electron–trion scattering in monolayer TMDCs in the low doping density limit. Our approach is fully microscopic with respect to all input parameters and functions, including matrix elements and the dielectric screening model. On the other hand, it is expected that the Fermi golden rule should break down at low doping densities close to the degeneracy crossover of the electron gas in the monolayer, and some caution must be exercised with respect to the use of the forms of the dielectric screening functions employed here.56 Avoiding these approximations allows for the description of a much broader range of doping but requires a full frequency-dependent many-body treatment.35,37,38

Accepting the above limitations, the calculations presented here still allow for some important conclusions to be drawn. First, we find that with a reasonable treatment dielectric environment, exciton linewidths arising from exciton–electron scattering on the order of 1 meV or higher are possible at low temperatures in the low-doping regime accessible in encapsulated, graphene-gated samples. Thus, even mild doping may provide a line broadening mechanism that can compete with (but not necessarily exceed) lifetime and phonon-related broadening in this regime. As expected from previous many-body calculations in the very low-doping regime, the growth of the excitonic linewidth is monotonic with increasing n, while the trion linewidth is largely insensitive to doping. However, we find that the trion linewidth is sensitive to temperature variations even over the small range T = 5–25 K, a somewhat unexpected feature from the standpoint of many-body theories such as that of Ref. 38 where the trion linewidth is partly described by a phenomenological input parameter. Lastly, we find that excited state exciton line broadening is somewhat larger and shows more sensitivity to increases in doping levels. Future work should be devoted to testing the veracity of these predictions and to understand how they merge with many-body approaches that have been applied to study the higher doping density regime.68 

In conclusion, we have provided a microscopic model for understanding how the scattering of excitons and trions surrounded by an electron gas in monolayer TMDCs may induce line broadening in the very low-doping density limit at low temperatures. A more detailed effort aimed at placing these contributions in the context of other mechanisms, such as exciton–phonon scattering, is worthy of future study. In addition, the approach adopted here may be of use for the calculation of the rates of processes such as Auger recombination69–71 in dimensionally confined systems. These and related topics will be the subject of future investigations.

The data that support the findings of this study are available from the corresponding author upon reasonable request.

M.R.C. acknowledges support from the United States Department of Energy through the Computational Sciences Graduate Fellowship (DOE CSGF) under Grant No. DE-FG02-97ER25308. D.R.R. acknowledges support by Grant No. NSF-CHE 1464802. The authors acknowledge fruitful discussions with Ian S. Dunn, Timothy C. Berkelbach, Guy Ramon, Archana Raja, Alexey Chernikov, and Mikhail Glazov.

Following the definition in Stern,49 in this appendix, we present the frequency-dependent 2D electron gas polarizability, χ, and its ω → 0 limit. The general form of χ is χ(q, ω) = χ1(q, ω) + 2(q, ω), where

χ1(z,ũ)=e2m2q2π1C(z,ũ)21ũ2z2C+(z,ũ)21+ũ2z2
(A1)

and

χ2(z,ũ)=e2m2q2πD(z,ũ)2z21ũ2D+(z,ũ)2z21+ũ2,
(A2)

where zq/2kF and ũ ≡ 2ωm/ℏq2. Note that the quantities in the braces, {·}, are dimensionless. The functions C and D are defined as follows:

C±(z,ũ)(z±ũ)/|z±ũ|if|z±ũ|>10otherwise
(A3)

and

D±(z,ũ)0if|z±ũ|>11otherwise.
(A4)

In the static approximation, we note that χ2(q, 0) = 0 and χ1 reduces to (15), where generally

ε2(q,ω)=2πB(q,ω)χ(q/2kF,2ωm/q2),
(A5)

and B(q,ω)=q2ε0ω2c2.

In this appendix, we outline the details of the X + eX + e scattering calculation, including accounting for electron spin. Here, and in  Appendix C, we closely follow the approach of Ref. 43, generalizing to the strict 2D limit and filling in necessary details.

In the following, it will be useful to keep in mind the electron and hole anti-commutation relations

{dks,cks}={dks,cks}={dks,cks}={dks,cks}=0

(electrons and holes always anti-commute) and

{xks,xks}={xks,xks}=0;{xks,xks}=δkkδss,

where x = c, d. Moreover, recall that ψke ends up as a global phase factor in the expression for the scattering rate and will be ignored in the following derivations.

1. General form of the matrix elements

A prudent first step to computing (5) is to split up V into its constituent parts and evaluate them independently on the initial state kxα,keβ, where spin indexes have been added as superscripts (the exciton spin references the electron; hole spin will not be important). In the case of the electron–hole component, we have

Vehkxα,keβ=kqvqϕ*ck+qαdkx+kqself-interactionckeβ0kqvqϕ*ckαdkx+kqcke+qβ0,
(B1)

where ϕ*ϕαxkx+k*. The first term in the above equation only contains information about the exciton interacting with itself (self-interaction) and is therefore discarded. The electron–electron component is calculated in a similar fashion and does not contain self-interaction terms,

Veekxα,keβ=kqvqϕ*ckqαdkx+kcke+qβ0.
(B2)

From here, by direct computation, we find the general matrix elements of the electron–exciton elastic scattering process to be

(kx+q)θ,(keq)ω|Veh|kxα,keβ  =kkqϕ1ϕ2*vq(δk,kθαδq,qωβδk,ke+qθβδk,keqωα),
(B3)

where ϕ1ϕ2*ϕαxkx+αxq+kϕαxkx+k*. Explicitly, this is

(kx+q)θ,(keq)ω|Veh|kxα,keβ  =vqδθαδωβkϕαxkx+αxq+kϕαxkx+k   ϕαxkxke+qδθβδωαkϕαxkx+αxqkekvk,
(B4)

which can be separated into direct (corresponding to vq) and exchange (vk) contributions. It is also observed that for practical computations, ϕ = ϕ* and thus the complex conjugation is dropped. The electron–electron term is computed as

(kx+q)θ,(keq)ω|Vee|kxα,keβ  =kkqϕ1ϕ2*vq(δk,k+qθαδq,qωβδk,ke+qθβδkq,keqωα)
(B5)

and simplified in a similar fashion,

(kx+q)θ,(keq)ω|Vee|kxα,keβ  =vqδθαδωβkϕαxkxβxq+kϕαxkx+kδθβδωα   ×kϕαxkx+αxqkekϕαxkx+qkekvk.
(B6)

Combining terms into direct and exchange contributions, we have

VD(q,ke,kx)=vqδθαδωβkϕαxkx+k[ϕαxkxβxq+kϕαxkx+αxq+k],
(B7)

where the Kronecker delta functions ensure that the proper spins are paired, and

VXC(q,ke,kx)=δθβδωαkvkϕαxqΔkx+k[ϕqΔkx+kϕqΔkx],
(B8)

where Δkxkeαxkx.

2. Spin contributions

Both the Vee and Veh terms can be split into clear direct and exchange contributions such that in the individual electron spin basis,

θω|V|αβ=δθαδωβVD+δθβδωαVXC.

If the incident and exciton electrons are in a singlet configuration, we have to consider all contributions from the singlet state S=()/2,

S|V|S=12(|V|+cc.|V|cc.),

which in the specified basis is

VSS|V|S=VDVXC.

By inspection, any of the triplet configurations are

VTT|V|T=VD+VXC.

In the case of the exciton case, the singlet and triplet contributions are essentially identical since the exchange contribution dominates, meaning |VS|2 ≈ |VT|2; for the trion, we do not consider triplet states.

3. 1s 1s scattering

With the assumption that the exciton wave function ϕ is in the parameterized ground state (1s) given by (3), the direct interaction has an analytic form. Noting that

kA(2π)2d2k,kR2,

and that the convolution

d2kg(λk)g(λ|q±k|)=2π(λ+λ)2gλλqλ+λ,
(B9)

the direct terms simplify to (dropping the spin Kronecker deltas)

V1sD(q)=2πe2Aqε(q)g(λβxq/2)g(λαxq/2).
(B10)

The exchange terms do not simplify and must be evaluated numerically,

V1sXC(q,ke,kx)=4e2λ2Ad2kkε(k)g(λ|αxqΔkx+k|)×g(λ|k1+qΔkx|)g(λ|qΔkx|).
(B11)

These results have been previously derived for scattering in finite quantum wells43 and match the results above in the 2D analytic limit. Here, λ = λ0 = 10.3 Å is the exciton effective Bohr radius, and αx = me/Mx is the mass ratio of the exciton, βx = 1 − αx, me = 0.49m0 and Mx = me + mh, where mh = 0.61m0.

4. 2s 2s scattering

To compute the excited state (2s) exciton elastic scattering linewidth, we parameterize a radial 2s hydrogen wave function,

ϕ2s(r,a,b)2rber/2a,
(B12)

in terms of an effective Bohr radius a and a secondary parameter b chosen to ensure orthogonality to the 1s state. An initial fit to the first excited state exact-diagonalization result of the Wannier exciton in a 2D Keldysh potential yielded length scales a = 7.79 Å and b = 5.33 Å, the latter of which was modified to b = 6.20 Å to ensure orthogonality. Fourier transforming to momentum-space yields

ϕk2s=N2s16πa2(1+4a2k2)3/22π12a2k2b14a2+k25/2
(B13)

with the normalization

N2s=b24πa2A(3a24ab+2b2).

Matrix elements are computed by making the substitution ϕϕ2s in (B7) and (B8) and numerically performing the 2D integrals.

The details of the T + eT + e scattering process are significantly more involved than the exciton case. The introduction of an extra electron manifests as another pair of creation and annihilation operators in the matrix element evaluation and adds many more terms. While the calculation is longer, it is no more conceptually difficult. In this appendix, we present the detailed derivation of the matrix elements for a 2D system, which coincide with the results for the 3D quantum well in the L → 0 limit.43 

The total elastic scattering matrix element V(q,ke,kt) is calculated by first computing the action of V|kt, ke⟩. This not only simplifies the number of operator contractions, but it also allows for the removal of self-interaction terms (those characterized by internal interactions between electrons and holes within the trion), as they do not contribute to the scattering matrix elements (similar to that of the exciton scattering case). To begin, we first evaluate the general contraction, which is used during the evaluation of (12),

c1c2c3c40=[δ12c3c4δ13c2c4+δ14c2c3]0,
(C1)

where in (C2), 1 ≡ (k1, z1, s′), 2 ≡ (−k1, z1, s1), 3 ≡ (−k2, z2, s2), and 4 ≡ (ke, ze, se), as this will be useful in computing both Veh|kt, ke⟩ and Vee|kt, ke⟩. In following calculations, hole operators will be ignored, as they do not contribute additional constraints or prefactors to the linewidth calculations. Moreover, Veh|kt, ke⟩ evaluates to

Vehkt,ke=k1,k2,k1,qs1,s2,se,svqξS*(s1,s2)Φαtkt+k1,αtkt+k2*ψke*ck1+qsck1sck1s1ck2s2ckese0=k1,k2,qs1,s2,seξS*(s1,s2)Φαtkt+k1,αtkt+k2*ψke*vqck1+qs1ck2s2ckeseck2+qs2ck1s1ckese+cke+qseck1s1ck2s20.
(C2)

The first two terms correspond to self-interactions between the internal electrons and holes of the trion. This is most easily seen by observing that after the action of Veh on the trion-free electron state, the initial incident electron momentum ke remains unchanged in the final creation operator. In the last term, however, we see that a momentum exchange of q′ has taken place.

The electron–electron terms corresponding to Veekt,ke are calculated similarly. As in the electron–hole case, we begin by performing the right-most contraction

c1c2c3c4c50=[δ15δ24δ14δ25]c30+[δ13δ25δ15δ23]c40+[δ14δ23δ13δ24]c50,
(C3)

where in (C3) and (C4) 1 ≡ (k2,s2), 2 ≡ (k1,s1), 3 ≡ (−k1, s1), 4 ≡ (−k2, s2), 5 ≡ (ke, se). In a similar fashion, Veekt,ke is thus found to be

Vee|kt,ke=12k1,k2,k1,k2,qs1,s2,se,s1,s2ξS*(s1,s2)Φαtkt+k1,αtkt+k2*ψke*vqck1+qs1ck2qs2ck2s2ck1s1ck1s1ck2s2ckese0=12k1,k2,qs1,s2,seξS*(s1,s2)Φαtkt+k1,αtkt+k2*ψke*vqck2+qs2ckeqseck1s1cke+qseck2qs2ck1s1+cke+qseck1qs1ck2s2ck1+qs1ckeqseck2s2+ck1+qs1ck2qs2ckeseck2+qs2ck1qs1ckese0=k1,k2,qs1,s2,seξS*(s1,s2)Φαtkt+k1,αtkt+k2*ψke*vqck1s1ck2qs2cke+qse+ck1qs1ck2s2cke+qse+ck1qs1ck2+qs2ckese0.
(C4)

In order to move from the second to the third equality in (C4), we have made use of the substitutions q′ → −q′ in the first, third, and fifth terms. The last term is a self-interaction exchange of momentum q′ between the two electrons on the trion.

With the self-interaction terms removed, and ignoring hole operators, the operation of V = Veh + Vee acting on the trion–electron scattering state is

Vkt,ke=k1,k2,qs1,s2,seξS*(s1,s2)Φαtkt+k1,αtkt+k2*ψke*vq×ck1qs1ck2s2+ck1s1ck2qs2ck1s1(z1)ck2s2cke+qse0.
(C5)

The trion–electron elastic scattering matrix elements are given by the action of kt+q,keq on (C5). Executing all possible integrals analytically produces a series of terms that can be broken into a direct component and two exchange components. We first adopt some notation: λ̃=λ1λ2/(λ1+λ2), is a harmonic-mean-like term that arises during convolutions, e.g., in (B9), and Δkt = keαtkt, where kt is the initial trion momentum and αt = me/Mt is the ratio of the effective electron mass to that of the trions Mt = 2me + mh. Finally, the elastic scattering matrix elements, V(q,ke,kt), are given by the sum of (C6) and (C8). The direct terms are

V D(q)=2πe2qε(q)A(1+κ2)j=15gj(q),
(C6)

where

g1(q)=g(λ1αtq/2)g(λ2βtq/2),g2(q)=g(λ2αtq/2)g(λ1βtq/2),g3(q)=2κ2g(λ̃αtq)g(λ̃βtq),g4(q)=g(λ1αtq/2)g(λ2αtq/2),g5(q)=κ2g2(λ̃αtq)
(C7)

and the exchange terms are

VXC(q,ke,kt)=2e2A(1+κ2)d2kkε(k)j=16Gj(q,k;λ1,λ2)+Gj(q,k;λ2,λ1),
(C8)

where

G1(q,k;λ1,λ2)=λ22g(λ1αtq/2)g(λ2|Δktq|)g(λ2|αtqΔkt+k|),G2(q,k;λ1,λ2)=λ22g(λ2|αtqΔkt+k|)g(λ1|αtqk|/2)g(λ2|Δktq|),G3(q,k;λ1,λ2)=λ22g(λ1αtq/2)g(λ2|qΔkt+k|)g(λ2|αtqΔkt+k|),G4(q,k;λ1,λ2)=κλ1λ2g(λ̃αtq)g(λ1|αtqΔkt+k|)g(λ2|Δktq|),G5(q,k;λ1,λ2)=κλ1λ2g(λ1|αtqΔkt+k|)g(λ̃|αtqk|)g(λ2|Δktq|),G6(q,k;λ1,λ2)=κλ1λ2g(λ̃αtq)g(λ2|qΔkt+k|)g(λ1|αtqΔkt+k|).
(C9)

Applying kt+q,keq on (C5) produces a series of integrals, many of which may be evaluated analytically. In addition, the signs, and in some cases the prefactor, of the various terms are determined by summing over the spin degrees of freedom.

As in previous calculations, it is helpful to evaluate the contraction of Fermionic operators

0|c1c2c3c4c5c6|0=δ16(δ34δ25δ24δ35)+δ15(δ24δ36δ34δ26)+δ14(δ35δ26δ25δ36),
(C10)

where for the electron–hole interaction,

1(keq,se),2(k2,s2),3(k1,s1),4(k1,s1),5(k2,s2),6(ke+q,se).
(C11)

As an example, the terms including δ16 produce g4(q) and g5(q) in (C7), and the remainder of the exchange terms correspond to G1 and G4 in (C9).

To evaluate the first (second) electron–electron interactions, we replace k1k1 + q′ (k2k2 + q′) in (C11). Each of the six terms generated by the contraction in (C10) is evaluated individually for the hole and two electrons, generating 18 total terms and producing (C7) and (C9). Note that g3(q) in (C7) accounts for two identical electron–electron interaction terms.

Instead of presenting a derivation of all 18 terms, we present a detailed derivation of one of them. The others follow similarly. Consider the electron–hole term corresponding to δ24δ15δ36,

12k1,k2,k1,k2,qs1,s2,se,s1,s2,sevqξS*(s1,s2)ξS(s1,s2)ψke*ψkeq  ×Φαtkt+k1,αtkt+k2*Φαt(kt+q)+k1,αt(kt+q)+k2  ×δk2,k1δs2,s1δkeq,k2δse,s2δk1,ke+qδs1,se.
(C12)

Note that the factor of 1/2 is due to an average over the initial free-electron spin states. We may sum over the dummy variables s1,s2,se,k2, k2 and q′. This yields

12k1,k1s1,s2,sevke+k1ξS*(s1,s2)ξS(se,s1)ψke*ψkeq  ×Φαtkt+k1,αtkt+qke*Φαt(kt+q)+k1,αt(kt+q)+k1.
(C13)

The spin factors are evaluated first,

s1,s2,seξS*(s1,s2)ξS(se,s1)=1ifs2=se0ifs2se.
(C14)

After sorting each term in (C13) by an integration variable and making the substitutions k1k1ke and k1k1αtkt, one integral may be evaluated analytically using the convolution in (B9), yielding G4(q, k′; λ1, λ2) after the substitution Δkt = keαtkt is made.

All integrations were performed using the Cubature adaptive integration package.72 Integrals over (0, ) were mapped to the finite range (0, 1) and performed using adaptive integration. Additionally, in order to avoid exhausting available memory, integrals were nested in the following way: First, scattering matrix elements were computed on the fly and converged to some relative error tolerance ϵ. This results in a computation of a 2D integral for the exchange terms. Once the matrix element V is computed, the golden rule integration that contains |V|2, along with the integration over all final states, is performed (this is a three-dimensional integral) and converged to some error tolerance , where c is typically on the order of 100–1000. Finally, to ensure convergence, the entire computation is converged with respect to the decrease in ϵ.

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