Intrinsic Exciton Linewidth in Monolayer Transition Metal Dichalcogenides

Monolayer transition metal dichalcogenides feature Coulomb-bound electron-hole pairs (excitons) with exceptionally large binding energy and coupled spin and valley degrees of freedom. These unique attributes have been leveraged for electrical and optical control of excitons for atomically-thin optoelectronics and valleytronics. The development of such technologies relies on understanding and quantifying the fundamental properties of the exciton. A key parameter is the intrinsic exciton homogeneous linewidth, which reflects irreversible quantum dissipation arising from system (exciton) and bath (vacuum and other quasiparticles) interactions. Using optical coherent two-dimensional spectroscopy, we provide the first experimental determination of the exciton homogeneous linewidth in monolayer transition metal dichalcogenides, specifically tungsten diselenide (WSe2). The role of exciton-exciton and exciton-phonon interactions in quantum decoherence is revealed through excitation density and temperature dependent linewidth measurements. The residual homogeneous linewidth extrapolated to zero density and temperature is ~1.5 meV, placing a lower bound of approximately 0.2 ps on the exciton radiative lifetime. The exciton quantum decoherence mechanisms presented in this work are expected to be ubiquitous in atomically-thin semiconductors.

residual homogeneous linewidth extrapolated to zero density and temperature is ~1.5 meV, placing a lower bound of approximately 0.2 ps on the exciton radiative lifetime. The exciton quantum decoherence mechanisms presented in this work are expected to be ubiquitous in atomically-thin semiconductors.
Monolayer transition metal dichalcogenides (TMDs) represent a new class of atomicallythin, direct bandgap semiconductors with coupled spin and valley pseudospin degrees of freedom (1,2). Fascinating phenomena have emerged in these materials with quantum confinement of carriers and excitons in the ultimate two-dimensional limit. Notable examples include a nonhydrogenic exciton Rydberg series (3,4), electronic and valley coherent coupling (5,6), and carrier spin and valley pseudospin Hall effects (7,8). In this work, we bring the new aspect of coherent quantum dynamics of excitons to this exciting field of research.
The quantum dynamics of an exciton are characterized by two key parameters, illustrated in Fig. 1A. The first is the excited state population decay time T1 (or decay rate, ), arising from both radiative and nonradiative population relaxation. The second is the dephasing time T2 of the coherent superposition of the ground (|0⟩) and excited (|1⟩) states. T2 is inversely proportional to the exciton homogeneous linewidth , which is linked to population relaxation through  = /2 + *, where * is additional broadening that arises from elastic, pure dephasing processes (9) such as exciton-impurity scattering. Exciton quantum dynamics can be probed in the frequency or time domains using optical spectroscopy. In practice, however, the individual exciton energies vary due to different local potentials arising from disorder and impurities, which inhomogeneously broaden the optical linewidth. Inhomogeneity conceals the intrinsic exciton homogeneous linewidth in many optical spectroscopy experiments (Fig. 1B).
Since the dephasing time sets the time scale over which excitons can be coherently manipulated, quantifying the source of exciton decoherence will provide essential information for the extensive efforts developing TMD-based optoelectronics, coherent valleytronics, and quantum information devices (10). For example, in semiconductor lasers the power dependent gain dynamics are sensitive to the interplay between homogeneous and inhomogeneous linewidths and cavity losses (11). In photovoltaic devices, dephasing may actually facilitate efficient exciton/energy transport analogous to the phenomenon observed in photosynthetic proteins (12) as illustrated in Fig. 1C. Systems with narrow homogeneous linewidth exhibit minimal energetic overlap between neighboring, quasi-localized excitons with different resonance frequencies, which inhibits efficient energy transfer. In contrast, strong dephasing often results in decoherence instead of transfer. Thus, the most efficient energy transfer is realized in systems with a delicate balance between disorder and decoherence (13).
Here, we investigate exciton coherent quantum dynamics in monolayer TMDs, specifically WSe2, using optical two-dimensional coherent spectroscopy (2DCS) (14). This technique unambiguously separates homogeneous dephasing from inhomogeneous broadening, revealing the intrinsic exciton linewidth. We show that the linewidth increases linearly with exciton density and temperature, a clear indication that exciton-exciton and exciton-phonon interactions play a substantial role in exciton decoherence. We extrapolate a zero-density, zero-temperature residual homogeneous linewidth of approximately 1.6 meV. This value is nearly two orders of magnitude smaller than the inhomogeneous linewidth, and it places a lower bound of 0.2 ps on the exciton radiative lifetime.
We examined monolayer WSe2 flakes ~10 µm in lateral size grown on a sapphire substrate using chemical vapor deposition (15, 16). In WSe2, two layers of selenium atoms separated by a layer of tungsten atoms form a 6-7 Å-thick hexagonal lattice ( Fig. 2A). The valleys of energymomentum dispersion appear at the ±K points of the first Brillouin zone. Strong spin-orbit coupling and time-reversal symmetry lead to large separation of the valence band states (17) and coupled spin-valley degrees of freedom at the ±K points. In monolayer TMDs optical selection rules for exciton states at the ±K valleys have led to optical control of carrier spin and valley pseudospin degrees of freedom (6,18). Here, we focus on the optical properties of the lowest energy transition corresponding to the A exciton of one helicity. The exciton resonance is first identified from the photoluminescence spectrum at ~10 K shown in Fig. 2B by the solid curve.
The spectrum features two peaksone corresponding to the exciton (X) at ~1700 meV and the other from localized, defect-bound excitons (D 0 X) at ~1650 meV (16,19). The full-width at halfmaximum (FWHM) of the exciton peak (in ≈ 50 meV) is determined by the inhomogeneous broadening as confirmed by 2DCS experiments presented below. Inhomogeneity can be ascribed to disorder potentials from defects such as chalcogenide vacancies and other impurities.
The homogeneous linewidth is measured using optical 2DCS, which is an extension of three-pulse four-wave mixing (Fig. S2) (20). Three phase-stabilized pulses separated by delays 1 and 2 coherently interact with the sample to generate a nonlinear signal field, ES(1,2, 3), that is emitted as a photon echo during a third time 3 (Fig. 2C). ES is phase resolved through spectral interferometry with a fourth phase-stabilized reference pulse, ER, yielding the emission frequency, of ES. The measurements are repeated as delay 1 is varied and subsequent Fouriertransformation of the signal field with respect to 1 generates a two-dimensional spectrum with amplitude given by ES(1,2, ). The absolute value of ES is shown in Fig. 3A for co-circular polarization for all pulses, a sample temperature of ~10 K, and an exciton population density of NX = 1.3×10 11 excitons/cm 2 (16). The ℏ1 axis is plotted as negative energy because the system evolves during 1 with the opposite phase accumulation relative to that during the detection time 3a result of the photon echo time-ordering of the pulses. The spectrum features a single peak on the diagonal line along ℏ3 = -ℏ indicating that the system coherently evolves with the same frequency during 1 and 3.
The inhomogeneous exciton energy distribution appears as a continuous elongation along the diagonal. In the present experiments the diagonal linewidth is limited by the laser bandwidth and does not reflect the amount of inhomogeneous broadening as determined from the photoluminescence spectrum. In contrast, the intrinsic dephasing rate of an individual exciton resonance is manifest as the width of the cross-diagonal lineshape along ℏ3 = ℏ, which is shown as the dashed line in Fig. 3B for an exciton resonance at 1710 meV. In the limit of strong inhomogeneity, as seen here, the homogeneous lineshape is well-described by the square root of a Lorentzian function with a FWHM equal to 2 (21). A least-squares fit to the data in Fig. 3B yields  = 2.7 ± 0.2 meV corresponding to an exciton coherence time T2 = ℏ/= 240 ± 20 fs. We present in Figs. 3C and 3D a two-dimensional spectrum and homogeneous lineshape, respectively, for an increased excitation density of NX = 1.3×10 12 excitons/cm 2 . The linewidth has increased by a factor of twoa clear signature of excitation-induced dephasing from interactions between excitons (22). Transient absorption measurements of the incoherent exciton population dynamics in WSe2 reveal that the exciton lifetime, T1, is independent of excitation density over the range used in these experiments (23). This result implies that excitation-induced dephasing arises from elastic exciton-exciton scattering that disrupts the phase coherence without transfer or relaxation of the exciton population under current experimental conditions.
The excitation density dependence of the homogeneous linewidth is shown in Fig. 4A by the solid points. Following a similar analysis performed for quasi-2D quantum wells (24,25), excitation-induced dephasing can be captured by the following expression: where 0 is the zero-density linewidth and * is an exciton-exciton interaction parameter. A fit of Eq. (1) is shown by the solid line in Fig. 4A, which yields * = 0.21±0.05×10 -11 meV cm 2 exciton -1 and an extrapolated zero-density homogeneous linewidth of  = 2.3±0.3 meV. The interaction parameter   is the same order of magnitude as in quasi-2D semiconductor quantum wells (22,(26)(27)(28). This is a nontrivial result, since a similar * would suggest stronger interaction broadening between tightly-bound excitons in monolayer TMDs (Bohr radius ~1 nm) compared to weaklybound excitons in quantum wells (Bohr radius 5-10 nm). Indeed, stronger exciton-exciton interactions in TMDs is consistent with reduced dielectric screening of the Coulomb force in atomically-thin materials (29,30).
We further examine the role of phonons in exciton decoherence by repeating the linewidth density dependent measurements as a function of temperature. We show the extrapolated zerodensity linewidth for temperatures up to 50 K in Fig. 4B. The linewidth increases linearly from 1.9±0.3 meV at 5 K to 4.5±0.3 meV at 50 K. The linear temperature dependence in this temperature range is reminiscent of exciton dephasing in semiconductor quantum wells due to absorption of an acoustic phonon with energy much smaller than kBT, where T is the sample temperature (31). Single-phonon anti-Stokes scattering can be modeled by where ph denotes the exciton-phonon coupling strength and 0(0) is the residual exciton homogeneous linewidth in the absence of exciton-exciton and exciton-phonon interactions. A fit of Eq. (2) to the data (solid line in Fig. 4B) yields ph = 60±6 eV/K, which is a factor of 5-10 larger compared to quasi-2D semiconductor quantum well systems (26,31). This value is also twice as large compared to bulk TMD InSe, in which optical phonons were shown to also contribute to low temperature (< 60 K) exciton dephasing (32).
The residual homogeneous linewidth (~1.6 meV) after removing exciton-exciton and exciton-phonon interaction effects can be ascribed to exciton population relaxation (T1 processes) and pure dephasing associated with impurities or defects (T2* processes). Fast population decay has been recently measured in WSe2 using time-resolved photoluminescence and transient nonlinear spectroscopies (19). In the sample investigated, a large number of impurities and defects are present as evidenced by the appearance of the impurity-bound exciton peak in the photoluminescence spectrum and the large exciton inhomogeneous linewidth (~50 meV). Thus, the residual homogeneous linewidth in the present sample can be attributed at least partially to pure dephasing. Such fast pure dephasing might arise from a fluctuating electrostatic environment due to charge capture events that Stark-shift the exciton energy on a picosecond timescale (33).
The homogeneous linewidth extrapolated to zero temperature and zero excitation density   The 2DCS experiment is performed using three phase-stabilized pulses separated by time delays 1 and 2 to coherently excite the sample. The photon echo signal is phase resolved, which allows a numerical Fourier-transform analysis to generate a two-dimensional spectrum.

Fig. 3. (A)
The photon echo signal appears as a single peak in the normalized two-dimensional spectrum (absolute value), acquired using co-circularly polarized pulses and an excitation density of ~1.3×10 11 excitons/cm 2 . The peak is inhomogeneously broadened along the diagonal line connecting ℏ3 = -ℏ1, whereas the cross-diagonal lineshape provides a measure of the homogeneous linewidth, . A normalized homogeneous profile relative to the exciton resonance frequency, 0, is shown in (B). The half-width at half-maximum of a square root of Lorentzian fit function yields a homogeneous linewidth of  = 2.7±0.2 meV (T2 = ℏ/= 240±20 fs). (C) A twodimensional spectrum for an increased excitation density of ~1.3×10 12   We note that the temperature of the sapphire substrate was at ∼750 to 850 °C when the center heating zone reached 925 °C. The heating zone was kept at 925 °C for 15 min and the furnace was then naturally cooled to room temperature. The reaction yielded triangular shaped flakes of WSe2 with a base width of ~10 m. The thickness was measured using atomic force microscopy, with a representative image shown in Fig. S1A. A height profile along the dashed line is shown in Fig.   S1B, confirming the ~7 Å monolayer thickness of the flakes. (C) Absorbance determined from differential reflection measurements at 17 K using the laser spectrum, yielding a maximum 0.12 at the peak of the laser. The laser spectrum (blue curve) is overlaid for reference.
The photoluminescence spectrum shown in Fig. 2B of the main text was acquired using 532 nm laser excitation and a sample temperature of 10 K. The peaks are identified through polarization analysis. The peak at ~1700 meV exhibits some degree of linear polarization following linearly polarized excitation (data not shown), which can only be attributed to the neutral exciton as a consequence of generating a coherent superposition of exciton valley states (6). In contrast, the peak at ~1650 meV does not exhibit linear polarization. The ~50 meV energy separation between the peaks is nearly a factor of two larger than the charged exciton binding energy of ~30 meV relative to the exciton (6,19). We therefore attribute this peak to localized, defect-bound excitons, which is also consistent with the WSe2 photoluminescence peak assignment in Ref. (19).
To determine the absorbance of the exciton, defined as = 1 − − , we measured the fractional change in the excitation laser reflectance for a single monolayer flake relative to the substrate reflectance. The differential reflectance (R) is related to the absorbance of a material on a transparent substrate by (1) where ns = 1.76 is the sapphire refractive index. The absorbance is shown in Fig. S1C for the laser tuned to a similar wavelength used in the 2DCS experiments and a sample temperature of 17 K.
From these measurements we find an exciton absorbance  ≈ 0.12. to /300. Such stability permits Fourier-transformation of the data and allows for phase cycling of the pulse delays to minimize scatter of the excitation pulses into the spectrometer, enhancing the signal-to-noise ratio. Three of the pulses with wavevectors k1, k2, and k3 are focused to a single 35 m spot FWHM on the sample (Fig. S2), which is kept at a temperature of 10 K in a liquid helium cold-finger cryostat. The first pulse, labeled E1 with wavevector k1 in Fig. 2D, generates an electronic coherence between the exciton ground and excited states. During the delay 1, the individual exciton resonances oscillate out of phase and the macroscopic coherence decays at a rate that is inversely proportional to the inhomogeneous broadening (or in the case of our experiment, the pulse spectral bandwidth). Upon the arrival of field E2 with wavevector k2, the electronic coherences are converted into a transient population grating. After a delay 2, field E3 with wavevector k3 generates a coherence whose phase evolution is reverse of that generated by field E1, resulting in a rephasing of the individual frequency components of the inhomogeneously broadened system. The coherent interaction of the three fields with the sample generates a thirdorder nonlinear optical signal field, ES (1,2,3), which is a photon echo that is detected in transmission in the wavevector-matching direction ks = -k1 + k2 + k3. ES is interferometrically measured using a fourth phase-stabilized reference field ER as the delay 1 is varied. Subsequent Fourier transformation of the signal field yields a rephasing two-dimensional spectrum with amplitude given by ES(1,2,3). We use a value of 2 = 0 fs to obtain maximum signal-to-noise; however using a value of 2 = 200 fs, which is larger than the pulse autocorrelation duration, does not result in any noticeable difference in the data other than an overall weaker signal strength due to population decay. Fig. S2. Box geometry for the 2DCS experiment. Three phase-stabilized pulses with wavevectors k1, k2, and k3 coherently interact with the sample to generate a photon echo radiated in transmission in the wavevector phase-matching direction, ks. The emitted signal field is measured through spectral interferometry with a phase-stabilized reference pulse.

Two
The excitation density can be calculated through the expression where Pave is the average power per beam, Tp = 12.5 ns is the laser pulse time separation, R = 0.15 takes into account reflection losses,  = 1e -L = 0.12 is the linear absorbance of the WSe2 monolayer, r = 17.5 m is the focused beam radius, and Eph = 1710 meV is the photon energy.

Microscopic calculation of the radiative lifetime of delocalized excitons
Starting with Maxwell equations and solving the wave equation for two-dimensional TMDs, we calculate the frequency-dependent excitonic absorbance ( ) = 1 − ( ) − ( ) with the transmission ( ) and the reflection coefficient ( ). Exploiting the boundary conditions for the electrical field for a two-dimensional TMD monolayer located between two media characterized by the refractive indices 1 and 2 , we obtain the following analytic expression for the absorbance (S1-S3) ( ) =  Figure   S3 shows the absorbance ( ) focusing on the A exciton. Our calculations reveal a homogeneous linewidth of 1.58 meV corresponding to a radiative lifetime of 208 fs. This value is consistent with the measurements and provides a lower bound on the exciton radiative lifetime.