Exciton-phonon coupling in MoSe monolayers
We study experimentally and theoretically the exciton-phonon interaction in MoSe monolayers encapsulated in hexagonal BN, which has an important impact on both optical absorption and emission processes. The exciton transition linewidth down to meV at low temperatures makes it possible to observe high energy tails in absorption and emission extending over several meV, not masked by inhomogeneous broadening. We develop an analytical theory of the exciton-phonon interaction accounting for the deformation potential induced by the longitudinal acoustic phonons, which plays an important role in exciton formation. The theory allows fitting absorption and emission spectra and permits estimating the deformation potential in MoSe monolayers. We underline the reasons why exciton-phonon coupling is much stronger in two-dimensional transition metal dichalcodenides as compared to conventional quantum well structures. The importance of exciton-phonon interactions is further highlighted by the observation of a multitude of Raman features in the photoluminescence excitation experiments.
Layered van der Waals materials provide a versatile platform for exploring fundamental physics and potential applications by combining different two-dimensional (2D) crystals Novoselov et al. (2016); Geim and Grigorieva (2013); Mak et al. (2010); Splendiani et al. (2010); Wang et al. (2012); Mak and Shan (2016). Among these layered materials, transition metal dichalcogenides (TMDs) are particularly suitable for fundamental and applied optics, as they are semiconductors with a direct bandgap when thinned down to one monolayer (ML) Splendiani et al. (2010); Mak et al. (2010). The Coulomb interaction between conduction electrons and valence holes is strong due to 2D confinement of charge carriers, heavy effective masses and weak screening. As a result, the optical properties of monolayer TMDs are dominated by excitons, electron-hole pairs bound by Coulomb attraction Wang et al. (2018); He et al. (2014); Ugeda et al. (2014); Chernikov et al. (2014); Ye et al. (2014); Qiu et al. (2013); Ramasubramaniam (2012); Wang et al. (2015a).
The interaction of excitons with phonons governs many important aspects of the optical properties of 2D materials Zhang et al. (2015); Ribeiro-Soares et al. (2014); Rice et al. (2013); Molina-Sánchez and Wirtz (2011); Horzum et al. (2013); Sekine et al. (1980); Jakubczyk et al. (2018) and semiconductor nano-structures in general Hameau et al. (1999); Lee et al. (1986); Tatham et al. (1989); Gindele et al. (1999); Urbaszek et al. (2004); Besombes et al. (2001); Borri et al. (2001) including energy relaxation, dephasing and transition linewidth broadening with temperature. Under non-resonant optical excitation high energy excitons are generated that subsequently loose energy by phonon emission, as recently discussed for MoSe MLs Chow et al. (2017a). Interactions of carriers with phonons play a key role in the exciton formation process Bonnot et al. (1974); Amand et al. (1994); Szczytko et al. (2004); Thilagam (2016). Other important signatures of the exciton-phonon interaction are single resonant and double resonant Raman scattering processes Guo et al. (2015); Carvalho et al. (2015); Wang et al. (2015b); Chow et al. (2017b); Molas et al. (2017); Soubelet et al. (2016). Also the possible impact of polarons, electrons dressed by a phonon-cloud, on the optical properties has been discussed Christiansen et al. (2017) for TMD MLs. There is a multitude of neutral and charged excitons in ML TMDs that give rise to complex photoluminescence spectra Koperski et al. (2017). Here phonons provide the necessary energy and in certain cases momentum for transitions between states in different valleys Dery and Song (2015); Dery (2016); Lindlau et al. (2017). Knowing details about the interaction of excitons with phonons is therefore important for exciton dynamics that dominates light-matter interaction in TMD MLs.
Recently encapsulation in hexagonal boron nitride (hBN) of TMD monolayers has resulted in considerable narrowing of the exciton transition linewidth, down to about 1 meV Jin et al. (2016); Manca et al. (2017); Chow et al. (2017b); Cadiz et al. (2017); Ajayi et al. (2017); Wang et al. (2016). This gives now access to interesting details of the exciton spectra. Here we show in photoluminescence excitation spectroscopy on monolayer MoSe encapsulated in hBN at T K a strong absorption tail at energies as high as 15 meV above the neutral exciton peak, well outside the transition linewidth. We demonstrate in resonant absorption experiments with a tunable laser that this high-energy tail is a very prominent feature in several samples investigated, underlining the importance of phonon-assisted exciton formation. In temperature dependent PL experiments we show a high energy emission tail appearing as the temperature is raised from 9 to 70 K. We argue that the efficient interaction between exciton and acoustic phonons is responsible for both the broadening in absorption and the unusual evolution of the PL line with temperature observed on the same sample. Based on model calculations within an analytical approach and taking into account the deformation potential interaction between excitons and acoustic phonons, we provide a fit of our absorption and emission experiments, discuss the role of phonons in exciton formation and compare our findings to exciton-phonon interactions in quasi-2D quantum well structures.
The paper is organized as follows: In Sec. II we report on experimental methods and results. Section III presents the model and analytical results. In Sec. IV we compare the experiment and theory and estimate the exciton – acoustic phonon coupling strength. Conclusions are given in Sec. V.
Ii Optical Spectroscopy Results
ii.1 Experimental methods
The samples are fabricated by mechanical exfoliation of bulk MoSe (commercially available from 2D semiconductors) and very high quality hBN crystals Taniguchi and Watanabe (2007) on 80 nm-thick SiO on a Si substrate. The experiments are carried out in a confocal microscope build in a vibration free, closed cycle cryostat with variable temperature K. The excitation/detection spot diameter is . The monolayer is excited by continuous wave (cw) Ti-Sa laser (Msquared solstis) with sub-eV linewidth. The photoluminescence (PL) signal is dispersed in a spectrometer and detected with a Si-CCD camera. The typical laser excitation power is 3 .
ii.2 Experimental results
In PL experiments shown in Fig. 1a we observe well-separated ( meV) emission lines for neutral (X at eV) and charged excitons (trions, T, at eV) not (). The transitions at low temperature in encapsulated MoSe MLs show very narrow lines with a linewidth down to 1.2 meV Cadiz et al. (2017). This gives access to physical processes previously masked by inhomogeneous contributions to the linewidth. We study now in detail quasi-resonant and resonant laser excitation of the X transition. We start by discussing PL excitation (PLE) experiments detecting the X PL emission intensity as function of excitation laser energy.
The integrated intensity of the X transition is plotted in Fig. 1b. We tune the laser from 1.651 eV to 1.747 eV in 2.5 meV steps (wavelength is incremented by nm). We observe oscillations of the PL intensity as a function of excitation energy, see also Fig.2a,c for different samples. The period of the oscillation is meV and exactly matches the longitudinal acoustic phonon energy at the M point of the 2D hexagonal Brillouin zone, LA(M), as discussed in detail by Chow et al. (2017a). As the excitation laser energy approaches the X transition, we make an important observation: in Fig. 1b the PL emission intensity starts to increase considerably for excitation energies below 1.653 eV, even though the X resonance is still 15 meV away and the X PL emission line is spectrally very narrow. We verified this surprising result by performing PLE measurements on different samples of MoSe MLs encapsulated in hBN, compared in Fig. 2.
As we approach the X transition with the laser, scattered laser light starts to obscure the PL signal on the CCD. In order to avoid this complication, we change detection scheme: We approach the X resonance further with our laser but we detect the trion emission and assume that it is a reasonable measure of X absorption. The excitation energy is tuned from 1.632 eV to 1.647 eV. The integrated trion emission intensity as a function of laser energy is recorded in Fig. 1c. This X absorption spectrum strongly deviates from a symmetric Lorentzian lineshape, with a pronounced tail on the high-energy side of the main resonance. We have repeated this experiments for different laser powers and have reproduced each time this highly asymmetric lineshape.
In addition to this high energy tail in absorption, we also obtain a surprising result when studying emission at different temperature. In temperature dependent PL from 10 K to 70 K of the X, shown in Fig. 1d, we observe an asymmetric broadening, with stronger broadening on the high energy side, a clear deviation from a standard Lorentzian emission lineshape.
In the following section we will explain the observed asymmetry in absorption and emission due to the efficient coupling of excitons to acoustic phonons. Our theoretical investigation is further motivated by the efficient exciton-phonon coupling observed for two additional samples in Fig. 2. Also for these sample we confirm the strong absorption tail at higher energy, well above the X resonance. In Figs. 2a and 2c for samples 2 and 3, respectively, this absorption tails is clearly observable up to 25 meV above the X resonance. At higher excitation energy also for these samples we observe the multi-phonon LA(M) resonances as for sample 1 in Fig. 1b.
Iii Theory of exciton-phonon interaction
The aim of this section is to develop a simplified but analytical approach to exciton-phonon interaction in TMD monolayers. Here we argue that taking into account emission and absorption of acoustic phonons the broadening of the exciton line both in absorption at low temperature shown in Fig. 1c and in emission at higher temperature shown in Fig. 1e can be explained. We can fit the experiments to demonstrate order of magnitude agreement between our model and the measurements. For a complementary, self-consistent approach the reader is referred to Christiansen et al. (2017), where a polaron framework was used.
iii.1 Deformation potential interaction
In semiconductors usually two types of interaction between excitons and acoustic phonons are discussed: the deformation potential interaction and the piezoelectric interaction Gantmakher and Levinson (1987); Car (). Although the latter is typically dominant for individual charge carriers Gantmakher and Levinson (1987); Car (); Danovich et al. (2017), the piezoelectric coupling is expected to be strongly suppressed for excitons in transition metal dichalcogenides due to close values of effective masses of electrons and holes (predicted by theory Li et al. (2013)) and also due to thesmall exciton Bohr radius.
The deformation potential interaction involves, in the axially symmetric approximation, longitudinal acoustic phonons. Hereafter we consider only the phonons from the MoSe monolayer because the electron-phonon deformation potential interaction occurs in general at short range. The van der Waals interaction between the monolayer and surrounding is relatively weak and, unlike ionic or covalent bonding, is not expected to efficiently transfer deformation from one layer to another. The exciton perturbation due to the deformation potential reads
Here is the phonon wavevector, () is the phonon creation (annihilation) operator, is the two-dimensional density of mass of the TMD ML, is the phonon frequency, is the (longitudinal) sound velocity, is the normalization area, () are the deformation potential constants for the conduction (valence) band, stands for the complex conjugate. The matrix element describing the exciton scattering from the state with the center of mass wavevector to the state with the center of mass wavevector accompanied by the phonon emission is derived from Eq. (1) with the result
where the form-factor
is introduced. Here and in what follows we assume equal electron and hole effective masses, is the ground state envelope function (taken in the hydrogenic form), is the effective exciton Bohr radius. For relatively small phonon wavevectors the form-factor is not sensitive to the shape of the envelope function.
iii.2 Exciton-phonon scattering
In this subsection we study the phonon-induced broadening of exciton resonance in the simplest possible approach based on the Fermi golden rule. To that end we introduce the out-scattering rate from the exciton state with the wavevector and kinetic energy , where is the translational mass of the exciton, due to emission of the acoustic phonons:
For , where , the exciton-phonon scattering is quasi-elastic Gantmakher and Levinson (1987) (this point is discussed in detail below in Sec. III.3), thus we can omit the phonon energy, , in the energy conservation -function. Assuming that typical phonon and exciton wavevectors are smaller than the inverse Bohr radius, (this condition is fulfilled at all realistic temperatures), we put the form-factor and obtain from Eq. (4):
Since exciton-phonon scattering is quasi-elastic both absorption and emission processes are possible because the phonon energy is typically much smaller than with being the temperature and the Boltzmann constant. Thus, for a thermalized exciton gas one has
is the phonon occupation numbers. For we have and, as it follows from Eq. (7)
The parameter can be accessed in experiments Selig et al. (2016); Jakubczyk et al. (2016); Cadiz et al. (2017), see also this work, Appendix A. Since the material parameters and the speed of sound are known to a large extent, the experimentally determined value of allows us to estimate the strength of the exciton-phonon interaction. Relevant parameters of exciton-phonon interaction are summarized in Tab. 1. The analysis in Appendix A demonstrates that the accuracy of determination of parameter is not extremely high because at moderate to high temperatures the broadening can be dominated by the optical phonons and acoustic phonons at the Brillouin zone edge. Therefore the values and the deduced deformation potentials can be considered as order-of-magnitudes estimates only.
iii.3 Phonon-assisted absorption and high-energy tail. Perturbative derivation
In this paragraph we aim to discuss exciton formation, assisted by phonons, for laser excitation energies below the first LA(M) resonance i.e. closer than 19 meV to the neutral exciton resonance. In absence of exciton-phonon interaction the absorbance of the transition metal dichalcogenide monolayer at the normal incidence of radiation in the spectral vicinity of exciton resonance is described by the Lorentzian profile
Here is the incident light frequency, is the exciton resonance frequency, is the exciton radiative decay rate, is the non-radiative decay rate of the exciton. This expression can be derived within the non-local dielectric response theory Glazov et al. (2014); Ivchenko (2005) by calculating the amplitude reflection and transmission coefficients of the monolayer and evaluating the absorbance as . Hereafter we disregard the effects related with the light propagation in the van der Waals heterostructure Robert et al. (2018). The estimates performed for our structures show that the asymmetry in the shape of the related to the light propagation effects is negligible.
To simplify derivations we consider in a first approach in what follows the limit of , which allows us to take into account the light-matter interaction perturbatively. In this limit we can arrive at Eq. (11) making use of the Fermi’s golden rule and evaluating the rate of the transitions to the excitonic state with the in-plane wavevector as
Here is the squared matrix element of the optical transition to the state normalized per flux of the incident photons. The -function ensuring the energy conservation is broadened by non-radiative processes
The radiative broadening of the exciton resonance can be introduced as . Within second-order perturbation theory the phonon-assisted absorption [schematically shown in Fig. 3, panels (c) and (d)] is possible for via a two step process: First the exciton is created in the intermediate (virtual) state with and the emission of the phonon takes place as a second step. Correspondingly, the contribution to absorbance due to this two-step process reads
is the detuning between the photon energy and the exciton resonance energy.
It follows from the momentum conservation law that the phonon wavevector as the total linear momentum of the electron-hole pair is equal to the in-plane photon momentum, that is close to zero at the normal incidence of radiation. The energy conservation law therefore reads
Equation (15) can be readily solved to obtain the phonon energy in the form ()
For typical parameters eV and all energy is transferred to the recoil exciton, leaving the phonon with a small energy . An illustration of the exciton and phonon dispersions is shown in Fig. 3c. Summation over in Eq. (13) can be performed for any ratio of to with the result
where is the Heaviside -function and the function is defined as
For the experimentally relevant situation where the function and in Eq. (17) reduces to
Expression (19) describes the tail of absorption at , is given by Eq. (16), similarly to predicted and observed in Ref. Christiansen et al. (2017). Accounting for both the phonon emission and absorption processes gives rise to an additional factor of in Eq. (19).
For the purpose of direct comparison with experiment, it is convenient to obtain a model expression for the absorption spectrum which accounts for both the (no-phonon) direct and indirect (phonon-assisted) processes. To that end we employ the self-consistent Born approximation Ando and Uemura (1974) for the exciton-phonon interaction, see Appendix B for details, and arrive at
where the acoustic phonon scattering rate is:
In Eqs. (20) and (21) both phonon absorption and emission processes are included. Strictly speaking, the self-consistent Born approximation is not applicable for , because the smoothing of the Heaviside function should be also taken into account. The fully self-consistent approach such as developed in Ref. Christiansen et al. (2017) should then be applied. Here, the condition is relaxed that is why is present also in the denominator of Eq. (20). The results of calculation of are shown in Fig. 3a and fit very well the measured absorption spectra, they are discussed further in Sec. IV.
iii.4 Temperature-dependent photoluminescence
In the photoluminescence process the exciton formed non-resonantly relaxes in energy towards small -states. For a monolayer in free space the light cone is defined as , is the speed of light, and when the exciton can recombine by emitting a photon. In the absence of exciton-phonon interaction the photoluminescence spectrum has a simple Lorentzian form, cf. Eq. (11). The exciton-phonon interaction makes states outside the light cone optically active as well. For example, an exciton with the wavevector can decay by emitting a phonon with wavevector and in addition a photon along the monolayer normal i.e. . For experiments at finite temperature not only the phonon emission, but also phonon absorption processes become important.
In general calculation of the photoluminescence spectrum requires detailed analysis of the exciton energy relaxation and recombination processes Deych et al. (2007); Piermarocchi et al. (1996). Here, to simplify the consideration as we aim for a qualitative comparison with our experimental data, we assume that the excitons are thermalized and their occupation numbers are described by the Boltzmann distribution
where is the chemical potential of the exciton gas. The increase in temperature mainly results in occupation of higher energy excitonic states, which due to the quasi-elastic character of the exciton-phonon interaction gives rise to the high-energy wing in the photoluminescence which is so striking for the experiments shown in Fig. 1e. Denoting, as in Sec. III.3, by the energy of the emitted photon, by the energy of the exciton at , and by the detuning, Eq. (14), we arrive at the photon emission rate in the form [cf. (13)]
Equation (23) holds for , in the energy conservation -function the phonon energy is omitted. The calculation shows that the photoluminescence spectrum at the positive detunings related to the phonon-assisted emission takes the form
The coefficient in Eq. (24) is temperature and detuning independent. For given detuning the relative photoluminescence intensity drastically increases with temperature mainly due to the presence of excitons with higher energies.
At the same time, the photoluminescence spectrum at (negative detuning ) is practically unaffected by the temperature increase. Indeed, to emit a photon with energy significantly (by more that ) lower than a phonon with the energy should be emitted as well. The rate of this process is given by
Due to the energy conservation law the process is allowed for small absolute values of detunings only , see Fig. 3. Thus, for the experimentally relevant parameters the photoluminescence increase at negative detunings is not expected in the theory, in agreement with the observation of an asymmetric lineshape in Fig. 1e. Note, that this is somewhat different from the optical spectra in quantum dot structures where the exciton energy spectrum is discrete and exciton-phonon coupling is essentially inelastic. Thus, in quantum dots deviations from Lorentzian line-shapes with wings at low and high energy can appear Besombes et al. (2001); Urbaszek et al. (2004).
Figure 3a (same data as Fig. 1c) shows the X absorption measured via the integrated trion PL intensity with a very prominent high energy tail. The results of our model calculations using Eqs. (20) and (21) for the best fit parameters (given in the caption) are shown by the red solid line. For comparison, blue dashed curve shows the symmetric Lorentzian which describes the low-energy wing of the absorption (), but does not reproduce the high energy wing. In contrast, the full calculation fits the spectrum and reproduces the observed absorption asymmetry very well, giving a strong indication that including exciton-phonon coupling is important for describing the absorption process.
The free parameters in our fit are the radiative, , and nonradiative, damping of the exciton as well as the difference of the deformation potentials for the electron and hole, . The values used for meV is in order of magnitude agreement with results from time resolved spectroscopy reporting hundreds of fs to few ps lifetimes Pöllmann et al. (2015); Jakubczyk et al. (2016); Robert et al. (2016). The best fit value of of eV is somewhat larger than the literature and experimental data summarized in Tab. 1. The discrepancy can originate from the simplification of the model, particularly, the absence of the coupling with optical phonons as well as with phonons at the Brillouin zone edge, as well as from certain inaccuracy in determination of the deformation potentials, see Appendix A for a brief discussion of accuracy of fit of the temperature-induced broadening. Additionally, the presence of free charge carriers may also provide a contribution to the high-energy wing in absorption due to the possibility to relax the momentum conservation law. We also stress that the absorption (Figs. 1c and 3a) is detected via the photoluminescence of the trion, therefore, efficiency of the relaxation pathway may play a role as well.
Similarly to the high energy tail in the absorption, the calculations predict the same asymmetry in the photoluminescence emission spectrum. As described in Sec. III.4 this is because the excitons with wavevectors become optically active due to the phonon-assisted processes. The increase in the temperature gives rise to the increase of the occupancy of excitonic states at larger wavevectors. This is exactly what is observed in Fig. 1(e). Figure 3b shows the intensity of photoluminescence at three values of the positive detuning , , and meV (all on the high energy wing) as a function of temperature. The experimental data are in reasonable agreement with the calculation after Eq. (24) and indeed confirm the importance of the phonons in the photoluminescence emission.
Finally, we stress that the exciton interaction with acoustic phonons is particularly strong in TMD MLsand, generally, in two-dimensional crystals as compared with conventional quantum well structures such as GaAs/AlGaAs. This is because the density of states of two-dimensional phonons is greatly enhanced compared to that of the bulk phonons . This results in about an order of magnitude enhancement (see Appendix C for details and further discussion) and accounts for the observation of the high-energy tails in exciton absorption in TMD monolayers.
Our combined experimental and theoretical study sheds light on the strong impact of the exciton-acoustic phonon interaction in TMD monolayers on exciton formation and recombination. We interpret strong absorption above the exciton resonance in terms of phonon-assisted exciton formation. Asymmetric lineshapes with high-energy tails in emission are observed in temperature dependent experiments. Also this unusual emission lineshape is due to deformation potential exciton-acoustic phonon coupling, which is enhanced in TMD monolayers as compared to conventional quantum well structures, and can be fitted by our simplified analytical theory.
Acknowledgements.— We are grateful to A. Knorr for discussions. We acknowledge funding from ANR 2D-vdW-Spin and ANR VallEx, ITN Spin-NANO Marie Sklodowska-Curie grant agreement No 676108 and ITN 4PHOTON Nr. 721394, and Labex NEXT project VWspin. X.M. also acknowledges the Institut Universitaire de France. Growth of hexagonal boron nitride crystals was supported by the Elemental Strategy Initiative conducted by the MEXT, Japan and the CREST (JPMJCR15F3), JST. M.A.S. and M.M.G. acknowledge partial support from LIA ILNACS, RFBR projects 17-02-00383, 17-52-16020 and RF President Grant MD-1555.2017.2.
Appendix A Thermal broadening of exciton line
Figure 4 presents the dependence of the exciton linewidth in photoluminescence (measured as full width at half maximum) on the sample temperature. Two fits are performed with very close values of parameters according to the general expression Jusserand et al. (2015); Selig et al. (2016); Dey et al. (2016)
Here is the zero-temperature linewidth not related to the exciton-phonon interaction, the -linear term is due to the exciton-acoustic phonon scattering, Eq. (10), and the last term describes the interaction of excitons with optical phonons or acoustic phonons at the Brillouin zone edges, is the effective phonon energy. Reasonable fit quality can be achieved for the values of which differ by a factor of , compare Fit - 1 and Fit - 2 parameter sets in the figure caption.
Appendix B Nonperturbative derivation
For comparison with experiment it is instructive to derive the expression for the indirect absorption spectrum relaxing the condition . To that end we calculate the self-energy of the exciton with with account for the exciton-phonon interaction (allowing for, as before, the processes with the phonon emission only). In the self-consistent Born approximation it takes the form [cf. Ref. Christiansen et al. (2017)]
is the “dressed” and
is the bare retarded Greens function of the exciton in the state . In derivation of Eq. (27), (28) we assumed that the photon damping is negligible and disregarded the diagrams with the self-crossings
The phonon-induced contribution can be calculated similarly to derivation of Eq. (17) with the result
With account for finite temperature effects (both emission and absorption) the contribution can be recast as
with given by Eq. (16).
The absorbance by the exciton is given by
We also note that for arbitrary relation between and the radiative damping should be included only in the denominator of Eq. (33), see Eq. (11) and Ref. Ivchenko (2005). At the general expression (33) passes to the direct absorbance, Eq. (11). At Eq. (33) provides the indirect transitions contribution derived in Eq. (19). The photon self-energy is given by, up to a factor , by the dressed exciton Greens function.
It is interesting to note that the approach developed above can be also applied for excitons interacting with free carriers or static disorder, if any. For example, for exciton interacting with short-range impurities the in Eqs. (30) and (33) should be replaced by the Born scattering rate (proportional to the impurity density and the electron-impurity scattering rate) which is independent on the exciton energy.
Appendix C Comparison with conventional quasi-two dimensional semiconductors
The analysis shows that exciton-phonon interaction is quite strong in transition metal dichalcogenide MLs as compared with conventional GaAs-based quantum wells. For example the parameter in GaAs quantum wells Jusserand et al. (2015) is an order of magnitude smaller than in MoSe despite higher values of deformation potential and the fact that exciton interacts with bulk (3D) phonons in GaAs-case. Qualitatively, the enhanced exciton-phonon interaction in atom-thin crystals is due to the fact that for small wavevectors the phonon density of states is larger in 2D than in 3D. One can demonstrate this difference between TMD MLs and quantum well structures like GaAs/AlGaAs by determining the phonon scattering rate, as in Eq. (7) but in this case for two-dimensional excitons interacting with bulk (3D) phonons:
where is the bulk mass density (gmcm), is the normalization volume, and the form-factor accounts for the exciton size quantization along the quantum well growth axis.In Eq. (36) below the sum over is removed due to the momentum conservation law, , while the sum over is transformed to the integral in the three-dimensional case as
where is the in-plane wavevector of the phonon, is the in-plane angle of the phonon wavevector, and is its normal component. By contrast, in the case of exciton interacting with two-dimensional phonons, the last integral over is absent. As a result, for phonons with small energies , one has, instead of Eq. (7)
where is the lattice constant and is the well width (we took into account that and ). The parameter is indeed small for typical quantum well structures. Its appearance in Eq. (36) as compared with Eq. (7) follows from the simple argument: for bulk phonons the phonon density of states is .
- In Ref. Kaasbjerg et al. (2012) analogous expression, Eq. (25), was derived for the electron-phonon scattering.
- From Ref. Jin et al. (2014).
- Calculated from the data on bulk crystals as , where is the lattice constant for the 2H polytype.
- From Ref. Kaasbjerg et al. (2012).
- From Ref. Selig et al. (2016)
- This work
- From Ref. Dey et al. (2016)
- From Ref. Cadiz et al. (2017)
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- Fit to experimental data on (thermal broadening of exciton resonance) after Eq. (10).
- From experimental data in Ref. Plechinger et al. (2015) on strain tuning of optical resonances.
- From experimental data in Ref. Schmidt et al. (2016) on strain tuning of optical resonances.
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