Thermodynamic Equilibrium between Excitons and Excitonic

Letter pubs.acs.org/JPCL

Cite This: J. Phys. Chem. Lett. 2019, 10, 3637−3644

Thermodynamic Equilibrium between Excitons and Excitonic Molecules Dictates Optical Gain in Colloidal CdSe Quantum Wells Pieter Geiregat,*,†,‡ Renu Tomar,†,‡ Kai Chen,¶,§,∥ Shalini Singh,†,‡ Justin M. Hodgkiss,¶,§,∥ and Zeger Hens†,‡ †

Physics and Chemistry of Nanostructures, Ghent University, Gent 9000, Belgium Center for Nano and Biophotonics, Gent 9000, Belgium ¶ The MacDiarmid Institute for Advanced Materials and Nanotechnology, Wellington 6012, New Zealand § School of Chemical and Physical Sciences, Victoria University of Wellington, Wellington 6012, New Zealand ∥ The Dodd-Walls Centre for Photonic and Quantum Technologies, Dunedin 9054, New Zealand

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‡

S Supporting Information *

ABSTRACT: We show that optical gain in 2D CdSe colloidal quantum wells (CQWs) shows little saturation and coexists with exciton absorption over a broad range of excitation densities, in stark contrast with 0D CdSe colloidal quantum dots (CQDs). In addition, we demonstrate that photoexcited CQWs can absorb or emit light through the thermodynamically driven formation or radiative recombination of singlet excitonic molecules. Invoking stimulated emission through the molecule−exciton transition, we can quantify all of the remarkable gain characteristics of CQWs using only experimentally determined parameters, an advance that highlights a fundamental difference between multiexcitons in CQWs and CQDs. While strong confinement prohibits the dissociation of multiexcitons into separate excitons in 0D CQDs, excitons and excitonic molecules coexist in a 2D CQW at room temperature, with densities governed by an association/ dissociation equilibrium, not by state-filling. Our finding points out future directions to optimize stimulated emission by excitonic 2D materials in general.

E

colloidal quantum wells (CQWs) have been put forward as efficient optical gain materials,17,18 literature is inconclusive as to what exact electronic transitions are involved in stimulated emission. Interpretations include the involvement of excitons,19,20 otherwise unspecified two-exciton states,20−22 or excimers formed by charge transfer between adjacent nanoplatelets.23 This diverse set of gain models is in stark contrast with the understanding of optical gain in 0D CdSe-based quantum dots (QDs). For these systems, net stimulated emission is explained through state-filling, where the state with two electron−hole pairs in a QDroutinely called the biexciton stateleads to net stimulated emission. This premise leads to quantitative models that are able to capture the most important gain parameters.24,25 Recently, attempts have been made to extend the state-filling model developed for QDs to the description of stimulated emission by colloidal QWs.20 Such an approach, however, disregards the fundamental difference between multiexciton states in QDs and QWs. Because the QD volume is smaller than the volume dictated by the exciton Bohr radius, a biexciton in a QD is simply the combination of two electron− hole pairs, regardless of the attractive or repulsive Coulomb

xcitons (X) are composite quasi-particles consisting of an electron and a hole held together by Coulomb attraction. Electronic transitions involving free excitons exhibit a strong light−matter interaction, yet the small exciton binding energy ΔX of only 1−10 meV restricts excitonics in bulk semiconductors to cryogenic temperatures. In two-dimensional (2D) materials, the reduced dimensionality leads to a 4-fold enhancement of the exciton binding energy, hence the observation of excitonic transitions at room temperature in epitaxial quantum wells.1 In recent years, a new class of 2D materials has appeared that are characterized by a low-κ dielectric environment. Examples include exfoliated mono- and multilayers of 2D van der Waals solids (or transition metal dichalcogenides, TMDs) such as MoS2,2−5 layered perovskites,6,7 and colloidal nanoplatelets of II−VI and perovskitetype semiconductors such as CdSe and CsPbBr3.8−10 In these systems, the reduced dielectric screening enhances ΔX as compared to epitaxial quantum wells, which opens a pathway to create high-density exciton systems at room temperature.11 Studies highlighting the excitonic properties of these lowscreening 2D materials abound,2,12,13 and the numerous reports on optical gain and lasing attest to the promise of using these materials in optoelectronics.14,15 Among these low-κ 2D materials, II−VI nanoplatelets stand out due to their high quantum yield, large exciton binding energy, and excellent colloidal stability.8,16 While these © 2019 American Chemical Society

Received: June 4, 2019 Accepted: June 12, 2019 Published: June 12, 2019 3637

DOI: 10.1021/acs.jpclett.9b01607 J. Phys. Chem. Lett. 2019, 10, 3637−3644

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The Journal of Physical Chemistry Letters interaction between both pairs.25 The large lateral area of a QW, on the other hand, makes that two electron−hole pairs can exist as either two separate, dissociated and mobile excitons or a single, associated quasi-particle consisting of two bound excitons (see Figure 1a). Clearly, such a quasi-

recombination presents a conceptual leap forward in terms of the mechanistic understanding of multiexciton physics in colloidal 2D materials. For this study, we synthesized two batches of 4.5 monolayer (1.37 nm) CdSe CQWs, large CQWs measuring 34 × 9.6 nm2 and small CQWs measuring 15.4 × 5.7 nm2, using the procedure first proposed by Ithurria et al. (see Supporting Information section S1).8 By means of a specific quenching procedure, we avoided the formation of CQW stacks.16 This point is confirmed by the transmission electron microsopy (TEM) image of the large CQWs (see Figure 2a), which

Figure 1. (a) Schematic representation of (red, X) excitons as electron−hole pairs bound by Coulomb interaction with binding energy ΔX and (blue, M) an excitonic molecule as an adduct of two excitons with a binding energy ΔM. KX and KM are the wavevector of the X and M center-of-mass motion, respectively. Note that both the excitons and molecules occupy areas πaB2 much smaller than the total CQW area, hence making the fusion of 2X into M not obvious. (b) Energy dispersion of the exciton and molecule as a function of KX and KM. EG denotes the single-particle gap. Electronic excitations involving a transitions (0 → X) from the ground state to the exciton state and (X → M) from the exciton to the molecule have been indicated. Due to wavevector conservation, only vertical transitions are allowed, i.e., KX = 0 for 0 → X and KX = KM for X → M.

particlehenceforth called the excitonic molecule (M)will only form in the case of attractive exciton−exciton interactions and high exciton density. Moreover, the overall distribution of electron−hole pairs over excitons and molecules will be determined by a thermodynamic equilibrium. In the case of epitaxial 2D systems and TMDs, several authors have elaborated on this thermodynamically driven formation of molecules,26−29 and the formation of these bound quasiparticles at room temperature is considered nontrivial,30,31 even for high-binding-energy systems such as TMDs.2,26 Given the mobility of both X and M in-plane, expressed by an inplane wavevector K (see Figure 1b), also the photon-assisted internal transitions between X and M are nontrivial and different from the 0D case where translational momentum is not necessarily conserved. Here, we show that 2D excitons in 4.5 monolayer CdSe CQWs can associate to form stable excitonic molecules with a binding energy ΔM of ∼45 meV. In a given platelet, this strong and favorable X−X interaction leads to a sizable population of molecules at room temperature in coexistence with a set of excitons, even at low exciton densities. Using polarized femtosecond pump−probe spectroscopy, we demonstrate the singlet nature of the molecule state and demonstrate that the X → M transition has an absorption coefficient exceeding 106 cm−1, a figure attesting to the strong light−matter interaction involved in molecule-related transitions. Invoking stimulated emission through the M → X transition under the condition of a thermal exciton−molecule equilibrium, we achieve for the first time an internally consistent and fully quantitative description of the CdSe CQW gain spectrum and threshold, using only parameters derived from experiments. By providing a quantitative description of the disruptively large optical gain in excitonic 2D materials, the assignment of optical gain by CdSe colloidal QWs to stimulated emission by molecule

Figure 2. (a) Bright-field TEM image of the large CdSe CQWs. (b) (Bottom) Spectrum of (red line) the intrinsic absorption coefficient and (filled red) PL of the large CdSe CQWs. The position of the HH and LH exciton line is indicated. (Top) Deconvolution of the absorbance spectrum using LH and HH exciton and free carrier contributions. The thus estimated exciton binding energy ΔX = 193 ± 5 meV is indicated. (c) 2D delay time/wavelength map of Δμi after femtosecond photoexcitation with 400 nm (3.1 eV) pulses with an energy of 45 μJ/cm2. The contours (dashed) PA and (dotted) gain indicate areas where a photoinduced absorption is measured or where the bleach signal exceeds the absorbance of the unexcited sample, respectively. (d) 2D delay time/wavelength map of the PL after femtosecond photoexcitation with similar pump conditions as those in (c). The black line represents the steady-state PL spectrum.

features well-separated CQWs only. As such, we study henceforth the properties of well-isolated CQWs, and coupling processes, such as for possible excimer formation,23 cannot occur. In addition, the long-wavelength tail characteristic of light scattering by CQW stacks is absent in the absorbance spectrum of the large CQWs, which shows only the characteristic absorption lines of the heavy hole (HH) and light hole (LH) exciton; see Figure 2b. Note that Figure 2b depicts the spectrum of the intrinsic absorption coefficient μi, which we obtained by rescaling the measured absorbance spectrum (see section S2). As outlined in section S3, this spectrum can be decomposed into separate contributions of the exciton and free carrier absorption from the HH and LH states, respectively. In line with previous literature reports, this 3638


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The Journal of Physical Chemistry Letters yields a HH exciton binding energy of ΔX = 193 ± 5 meV; see Figure 2b.32,33 Figure 2c represents the delay time/wavelength map of the transient absorption coefficient Δμi of the large CQWs after photoexcitation using 400 nm, 45 μJ/cm2 femtosecond pulses, which create on average ⟨N⟩ = 7.5 excitations per CQW (see section S4). In line with previous reports, we found that Δμi is dominated by bleaches of the HH and LH transition.34,35 Similarly, the radiative recombination of the HH exciton is the main feature in the delay time/wavelength map of the photoluminescence (PL) shown in Figure 2d, which exhibits a similar decay as the HH bleach. Gauging the temporal exciton population by the PL intensity, we can describe this decay, in agreement with literature reports, as a mixed secondorder/first-order process (see section S5).36,37 More intriguing elements of the Δμi maps are the areas highlighted by the contours in Figure 2c at the red side of the HH bleach. Indicated as PA, a first involves a short initial burst of photoinduced absorption, which peaks 45 meV to the red of the HH exciton line. With increasing delay, this photoinduced absorption gives way for a long-lasting bleach that exceeds μi in absolute valueindicative of net stimulated emissionin the area indicated as gain. Interestingly, this gain band is mirrored by a transient broadening of the PL spectrum shown in Figure 2d. This regime strongly contrasts with the bleach of the HH exciton, which does not result in net stimulated emission. At even longer delay times, the photoinduced absorption again dominates, albeit in a more narrow wavelength range. As is established from work on epitaxial quantum wells at cryogenic temperatures, excitons in 2D semiconductors can associate to form a so-called excitonic molecule with a binding energy ΔM that should be 0.228 × ΔX in 2D quantum wells.38,39 This amounts to ∼43 meV in the case of the CdSe CQWs studied here, a value corresponding to the PA region in the transient absorbance map. Because a stable molecule should only be formed in a singlet state with total angular momentum F = 0 for zinc-blende CdSe,40,41 the possible formation of molecules can be verified through circularly polarized pump−probe spectroscopy.12 As indicated in Figure 3a, a σ+σ− sequence of right-handed pump and left-handed probe light can create molecules, and so does a σ−σ+ sequence, whereas σ+σ+ or σ−σ− sequences cannot. Figure 3b represents ΔA maps measured using σ+ and σ− probe light, after resonant excitation with 510 nm σ+ pump light; see the Methods section and section S6 for experimental details. It can be seen that the σ+σ− sequence reproduces the initial PA burst present in the unpolarized TA map, while the σ+σ+ sequence leads to only a gradual increase of the PA signal. We thus assign the PA region indicated in Figure 2c to light absorption linked to the X → M transition, where the decay of the counter-polarized and the buildup of the co-polarized transient absorbance can be ascribed to rapid spin−flips that redistribute the initial spinpolarized exciton state across all available exciton states; see section S6. As shown in Figure 3c, we obtain the absorbance spectrum of the X → M transition by a deconvolution of the counterpolarized transient absorbance recorded shortly after photoexcitation. In line with reports on epitaxial systems,28 we interpret the ensuing difference of 45 meV between the maximum of the resulting X → M spectrum and the maximum of the exciton bleach as the molecule binding energy ΔM. Additionally, this spectrum can be understood by considering the dispersion of the center-of-mass motion of the exciton and

Figure 3. (a) Diagram indicating the optical transitions between the different spin-polarized states of the exciton X and the molecule M. Dashed gray lines indicate the dark J= ±2 exciton states. (b) ΔA map recorded on the large CQWs after pumping at 510 nm, creating ⟨N⟩ = 8 excitations per platelet using (bottom) co-circular σ+σ+ and (top) counter-circular σ+σ− pump−probe polarization. (c) ΔA recorded at a delay of 250 fs using (blue) co-circular and (red) counter-circular pump−probe polarization. The spectrum of the X → M absorbance obtained after decomposing σ+σ− transient absorbance is explicitly (E) versus shown. (d) Integrated intrinsic absorption coefficient μX→M i excitation number. The linear scaling yields the absorption coefficient = 1.84 × 106 cm−1. μX→M i,0

the molecule; see Figure 1b. As outlined in section S7, the absorption coefficient μX→M (E) can be written as the product i of an intrinsic absorption coefficient μX→M of the transition and i,0 the probability f that the concomitant center-of-mass state is occupied. Referring to Figure 1b and section S7, we thus have μi X → M (E) = μi,0X→ M f (2(EM,0 − E)) = μi,0X→ M

π ℏ2 ⟨N ⟩ −2[(EM,0 − E)/ kBT ] e mX kBT S

(1)

Here, we replaced f by a Boltzmann distribution, denoted the exciton mass as mX, and used E for the probe photon energy. Phonon coupling will broaden this exponentially decaying spectrum to yield the measured X → M spectrum shown in Figure 3c. Therefore, μX→M is best obtained from the energy i,0 integrated spectrum. As indicated in Figure 3d, this integral scales proportionally to ⟨N⟩ with a slope yielding μX→M = 1.84 i,0 × 106 cm−1. Figure 4a represents a map of the material gain gi = −(μi + Δμi) of the large CQW sample. Here, the color coding makes only the region where gi > 0 show up, a condition corresponding to net stimulated emission. In this case, 400 nm pump light was used with a fluence of 90 μJ/cm2, which yields, on average, ⟨N⟩ = 15 excitations per platelet. In agreement with the Δμi map shown in Figure 2c, we found a transient gain band extending from 515 to 580 nm. The gain band narrows down and loses intensity with increasing delay and vanishes after ∼100 ps. Figure 4b shows TA and PL spectra of the large CQWs at a 3 ps delay, recorded for 3639


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CdSe CQWs and CdSe/CdS QDs have markedly different gain characteristics. In contrast to QDs, where state-filling models can successfully describe optical gain,24,25 multiple mechanisms have been put forward to describe optical gain in 2D systems such as quantum wells, which involve either free carriers,43 excitons,44 or, at cryogenic temperatures, excitonic molecules.30 Considering the high exciton binding energy and the overlap between the X → M absorption, the M → X spontaneous emission, and the net stimulated emission band, we start from the conjecture that optical gain in CdSe CQWs results from stimulated emission through the M → X transition. As shown in Figure 5a, the net absorption

Figure 4. (a) Time−wavelength map of the material gain gi after photoexcitation at 400 nm with a pulse energy of 90 μJ/cm2, a pulse creating, on average, ⟨N⟩ = 15 excitations per platelet. (b) Gain and PL spectra obtained at a 3 ps time delay for increasing pump fluences, characterized by the excitation number ⟨N⟩, as indicated. (c) Excitation number at which net optical gain is first observed, i.e., the gain threshold, for the (filled squares) large and (open circles) small CQWs. (d) Material gain gi,530 at 530 nm 3 ps after photoexcitation as a function of the average exciton density ⟨n⟩ for large and small CQWs.

different initial excitation numbers ⟨N⟩. It can be seen that the onset of net optical gain at ⟨N⟩ ≈ 5 concurs with the appearance of the red-shifted PL band that was already highlighted in the PL map shown in Figure 2d. As outlined in section S8, this long-wavelength PL band exhibits a superlinear increase with pump power. Given the spectral overlap of this emission with the X → M absorption, we therefore assign this band to radiative M → X recombination, an indication that also optical gain could be related to stimulated emission across the same M → X transition. A more detailed analysis of the gain threshold yields ⟨Nth⟩ = 4.1 for the large CQWs, whereas the small CQWs feature a threshold at ⟨Nth⟩ = 1.2 (see Figure 4c and section S9). Normalizing these thresholds by the CQW area, we find that both numbers correspond to the same threshold excitation density ⟨nth⟩ of 0.013 nm−2, which is attained at a threshold fluence of 25 μJ/cm2. Further, using ⟨n⟩ as a measure for optical gain in CQWs, Figure 4d represents the material gain gi,530 at 530 nm, recorded 3 ps after photoexcitation. It follows that gi is indeed a function of the excitation density ⟨n⟩. Moreover, even at excitation densities that are 10-fold of ⟨nth⟩, no saturation of gi occurs. As outlined in section S9, a further increase of the pump fluence makes gi,530 level off at around 1.5 × 104 cm−1, a figure exceeding the material gain of CdSe/ CdS core/shell QDs or bulk-like perovskite nanocrystals by almost 1 order of magnitude.24,42 Moreover, as gain in QDs is best measured using the excitation number N, not the density, and shows rapid saturation with increasing N, we conclude that

Figure 5. Quantitative optical gain model. (a) Stimulated emission (blue arrows) from molecules can overcome competing absorption (red arrow) at large values of the in-plane momentum K. (b) Populations of excitons ⟨NX⟩ and molecules ⟨NM⟩ and their ratio R = ⟨NM⟩/⟨NX⟩ as a function of total excitation number ⟨N⟩. An example case of ⟨N⟩ = 15 is highlighted. (c) Excitation number as extracted from time-resolved PL. Time delays of 20, 40, and 60 ps correspond to ⟨N⟩ = 36, 22, and 15, respectively. The inset shows the temperature of the exciton gas as extracted from the PL spectrum; see section S11. (d) Representation of the (cross markers) experimental and (solid lines) simulated material gain spectrum, showing excellent correspondence. The experimental spectra correspond to the delays indicated in (c), and the corresponding ⟨N⟩ and T were used in the simulation.

(E) of a mixed population of free excitons coefficient μX→M i and molecules amounts to the sum of X → M absorption and M → X stimulated emission. Considering the masses and the electronic degeneracies of the exciton and the molecule, this description yields μX→M (E) as (see section S11) i μi X → M (E) = μi,0X→ M

π ℏ2 ⟨NX⟩ −(EM ,0 − E)/ kBT e 4mX kBT S

× (e−(EM ,0 − E)/ kBT − 2R ) 3640

(2)


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The Journal of Physical Chemistry Letters Here, E is the photon energy and R denotes the ratio ⟨NM⟩/ ⟨NX⟩ between the molecule and the exciton number. Equation 2 indicates that the absorption coefficient of a given mixture of excitons and molecules will always become negative, however small R, at photon energies E sufficiently to the red of the molecule formation threshold EM,0. Referring to Figure 5a, this absence of an intrinsic lower limit for the gain threshold results from the smaller curvature of the molecules’ center-of-mass energy dispersion; see also Figure 1. To predict the material gain spectrum of CdSe CQWs using eq 2, EM,0, μX→M i,0 , ⟨N⟩, and R must be known. While pump− probe spectroscopy already yielded EM,0 and μX→M i,0 , we rely in what follows on an association/dissociation equilibrium between excitons and molecules to obtain R; see Figure 5b and section S10.27 This is a key assumption that differs fundamentally from state-filling models put forward to describe optical gain by CQWs.20 Indeed, state-filling identifies a state of two excitons in a CQW with a biexciton, a state of three excitons with a triexciton, and so on. A thermodynamic equilibrium, on the other hand, allows for the ⟨N⟩ = 2 exciton state to exist as two separate excitons or a single molecule, whereas the ⟨N⟩ = 3 state can correspond to three separate excitons, an exciton and a molecule, or, rather unlikely, a triexciton molecule. Restricting the bound states to excitons and molecules, the average number of both quasi-particles per CQW follows from a dissociation equilibrium27 2X F M

the equilibrium constant K scales with the surface area of the CQWs, the equilibrium model can be rewritten as a relation between the exciton and molecule density ⟨NX⟩/S and ⟨NM⟩/S as a function of the excitation density ⟨N⟩/S. Hence, eq 2 yields a net absorption or gain coefficient that is a function of the excitation density ⟨N⟩/S, rather than the excitation number ⟨N⟩, a point that can explain why, in the case of CQWs, the gain threshold corresponds to a fixed threshold density rather than an excitation number. Because eq 2 applies to only a mixed exciton−molecule gas in quasi-equilibrium, we focus on predicting material gain spectra recorded after longer delay times. In such cases, we can use the integrated instantaneous PL spectrum to track the excitation number ⟨N⟩ and the temperature of the exciton− molecule gas; see Figure 5c and section S11. Figure 5d displays three experimental gain spectra (markers)taken at different pump−probe delay times after a pump pulse, creating ⟨N⟩ = 55 excitationstogether with the gain spectrum as predicted by eq 2 (solid lines). As can be seen, the assumption underlying eq 2 that optical gain in CdSe CQWs results from M → X stimulated emission in a mixed population of 2D excitons and molecules leads, for the first time, to a fully consistent and quantitative description of the material gain spectra over a large range of excitation densities, without needing any ill-defined fitting parameter. This agreement indicates that the remarkable optical gain by CdSe CQWs is brought about by stimulated emission across the M → X transition, where the formation of the molecules is dictated by a thermodynamic equilibrium. Though existing literature on optical gain in CQWs seems to agree on the fact that multiexcitons are involved in the stimulated emission process, the current state-of-the-art provided no such insights in how to model the relative populations of excitons and multiexcitons nor on the formal and quantitative description of the stimulated emission cross section or, equivalently, the gain magnitude and the gain spectrum. The different optical gain characteristics of CdSe CQWs and QDs indicate that excitons in 2D CdSe CQWs behave very differently from excitons in 0D CdSe-based QDs. The approach underlying the molecule gain model indicates that the difference is derived from the fact that in 0D QDs the biexciton cannot dissociate to form two separate excitons. Because the QD volume is smaller than the exciton Bohr volume, two excitons in 0D QDs are forced to interact, regardless of the exciton binding energy being attractive or repulsive. Opposite from this, 2D excitonic molecules can readily dissociate in a CQW, whose lateral dimensions strongly exceed the 2D exciton Bohr area. As a result, the multiple electron−hole pairs created by photoexcitation condense to form a gas of excitons and molecules whose composition is determined by the thermodynamics of the exciton−molecule association/dissociation equilibrium. As outlined in section S12, this fundamental difference between excitons in QDs and CQWs accounts for the coexistence of optical gain and exciton absorption over a broad range of pump powers in CQWs, a finding state-filling models inevitably struggle to explain.20 Indeed, state-filling unavoidably predicts that the complete bleach of the exciton line gives way to net stimulated emission with increasing pump power. While this behavior is welldocumented for QDs, the relation between exciton absorption and optical gain in CQWs is different. The TA spectra shown here, see also section S12, and in the literature highlight that optical gain and exciton absorbance effectively coexist over a

(3)

This equilibrium requires that the number of excitons ⟨NX⟩ and molecules ⟨NM⟩ be related by the equilibrium equation (K: equilibrium constant) ⟨NM⟩ ⟨NX⟩2

=K (4)

Denoting the initial number of excitations per CQW as ⟨N⟩ = ⟨NX⟩ + 2⟨NM⟩, the average number of excitons and molecules per CQW can then be expressed as ⟨NX⟩ =

⟨NM⟩ =

8K ⟨N ⟩ + 1 − 1 (5)

4K 4K ⟨N ⟩ + 1 − 8K

8K ⟨N ⟩ + 1 (6)

Using the approach laid out by Gourley et al. and adopted by others, the equilibrium constant K can be expressed in terms of the translational and electronic partition functions of the exciton and the molecule, respectively (see section S10)27,45,46 K=

4π ℏ2S −ΔM / kBT e gX 2mX kBT

(7)

Note that the notion of mobile excitons and molecules, both showing translation degrees of freedom, is key to obtain this expression, again a fundamental difference from 0D excitons or biexcitons. Figure 5b represents the variation of ⟨NX⟩ and ⟨NM⟩ with increasing excitation number ⟨N⟩, calculated according to the approach outlined above and taking the degeneracy of the HH exciton gX = 4 and the molecule binding energy ΔM = 45 meV (see section S10). For an electronic temperature of 400 K, Figure 5b shows that large CQWs will hold one molecule when ⟨N⟩ = 7, an occupation only slightly above the gain threshold for such CQWs. Moreover, because 3641


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troscopy stand out. In combination with the versatile device integration, CQWs therefore offer a most feasible pathway to investigate, develop, and implement room temperature excitonics.

broad range of excitation densities in CdSe CQWs. In the case of the large CQWs, for example, optical gain first occurs when ⟨N⟩ ≈ 4, while the exciton absorbance persists up to at least ⟨N⟩s ≈ 100. If state-filling were to apply, the gain threshold should amount to Ns/2 ≈ 50 at least, a number that exceeds the actual threshold by 1 order of magnitude. A gain mechanism based on the M → X transition and thermodynamic formation of bound multiexciton molecules accounts for all of the attractive gain characteristics of CdSe CQWs. For one thing, the material gain is given by the product of the intrinsic absorption coefficient μX→M of the X → M i,0 transition, which we estimate for CdSe CDQs at 1.8 106 cm−1, and the probability that molecules occupy a given center-ofmass translational state. Because excitons and molecules are bosons, nothing restricts, in principle, the accumulation of multiple molecules in a single CQW, and hence, no rapid saturation of the gain coefficient is to be expected with increasing pump power. Hence, the possibility to attain an exceptionally high material gain of around 1.5 × 104 cm−1 at the highest excitation densities used in this work. These exceptional gain coefficients were also observed by other groups, yet no rational explanation could be given.18 The resulting combination of a high material gain and little gain saturation makes such CQWs ideally suited for realizing highpower, small-footprint lasers. From a materials perspective, the finding that excitons in CQWs associate to form molecules, with the M → X transition leading to optical gain, has several intriguing implications. First, we expect an increased molecule binding energy, e.g., in thinner II−VI CQWs or TMDs, or a reduced temperature to significantly enhance the material gain and further lower the gain threshold, the latter having no intrinsic lower limit. Moreover, the lifetime of the inverted state will eventually be limited by only the two-body recombination time, which can amount to several nanoseconds in CdSe CQWs; see section S5. In addition, the high stability of excitonic molecules suggests that these quasi-particles will be the dominant species at cryogenic temperatures. This can significantly modify the process of exciton condensation because the interparticle interactions of heavy molecules might differ significantly from their exciton counterparts.47,48 Finally, the ability to study stable excitons and excitonic molecules at room temperature opens up the possibility to investigate novel collective phenomena49 and higher-order complexes such as triexcitons without the need for cryogenic temperatures or complex coherent multiphoton interactions.50 In summary, this study builds on a series of findings that include the 20-year old observation of excitonic molecules and stimulated emission through the M → X transition for quantum wells at cryogenic temperatures30 and the recent demonstration that in monolayer TMDs stable excitonic molecules are formed with binding energies well above thermal energy at room temperature.2,12,26 The confirmation that colloidal CdSe quantum wells also host stable molecules and the conclusion that such molecules live sufficiently long to give rise to net stimulated emission demonstrate that low-screening 2D materials enable excitonic transitions to be used for optoelectronics at room temperature. In this respect, CQWs have multiple advantages. In particular, the atomically precise and controllable synthesis, solution processability, strong suppression of nonradiative exciton recombination thanks to a nearly perfect surface termination,16 and ideal fit between transparent nanocolloids and quantitative femtosecond spec-

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METHODS Synthesis of Colloidal CdSe Quantum Wells. We synthesized the batches of 4.5 monolayer CdSe CQWs or CQWs used in this study following the procedure originally developed by Ithurria et al.8 and modified by Singh et al.16 We started by degassing cadmium myristate (0.34 g), Se (24 mg), and octadecylether (ODE) (25 mL) in a three-necked flask using vacuum and backfilling with N2. Next, 0.08 g of cadmium acetate was added to this mixture at 205 °C when the solution turned yellow. Afterward, the reaction mixture was heated to 240 °C for 10 min to form the large CQW sample, after which the reaction was quenched by the addition of 1.6 g of cadmium oleate in ODE. We reduced the reaction time at 240 °C to 1 min to synthesize the small CQW sample. In both cases, samples were washed with a hexane/ethanol mixture. For spectroscopy, the CdSe CQWs were dispersed in spectroscopy-grade n-hexane at concentrations that yield an optical density of 0.1 at the first HH exciton transition across a 1 mm cuvette. Transmission Electron Microscopy. Bright-field TEM images were recorded with a Cs-corrected JEOL 2200-FS operated at 200 kV. Samples were drop-cast from n-hexane on copper grids. Sizing was done for at least 250 particles by calculating the rectangle-shaped cross section in the TEM images from different areas on the grid. UV−Vis Absorbance Spectroscopy. Absorbance spectra were measured with a Perkin-Elmer Lambda 950 UV−vis−NIR spectrophotometer. Steady-State Photoluminescence Spectroscopy. The quantum yield of the PL of the CdSe CQWs upon excitation with 480 nm light was determined using an integrating sphere and was found to be 52 ± 5% for the large CQWs and 45 ± 5% for the small CQWs. Transient Absorption Spectroscopy. To measure the transient absorbance, dispersions of CdSe CQWs in hexane were photoexcited using 110 fs (fs) pulses with varying wavelengths. These pump pulses were created from the amplified 800 nm fundamental of a Ti:sapphire laser (Spitfire Ace, Spectra Physics, 1 kHZ) through nonlinear frequency mixing in an optical parametric amplifier (TOPAS, Light Conversion). The broad-band probe was generated by focusing the 800 nm fundamental on a thin CaF2 crystal. The pulses were delayed relative to the probe using a delay stage with a maximum delay of 3 ns (Newport TAS). Noise levels of 0.1 mOD (RMS) were achieved by averaging over 10000 shots. The probe spectrum covered the vis−NIR window from 350 up to 750 nm, though our emphasis is on the 480−600 nm window. We used cuvettes with a 2 mm path length, and the dispersions were stirred during pump−probe measurements to avoid effects of photocharging and sample degradation. No specific air-free sample handling was required as the CdSe CQWs proved insensitive to oxidation, as evidenced, for example, by a lack of PL peak shift or change in PL quantum yield when exposing samples to ambient conditions and/or the laser pulses. To obtain circular polarization for the pump and probe, a combination of linear polarizers and (broad-band) quarterwave plates with high polarization contrast was used. Ultrafast Luminescence Spectroscopy. For femtosecond PL spectroscopy, dispersions of CdSe CQWs were loaded in a 3642


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travel grant. J.M.H. and K.C. acknowledge support from the Marsden Fund.

cuvette with a 1 mm optical path length and an absorbance of 0.2 at the HH absorption peak to avoid strong reabsorption. Samples were translated along one axis to avoid photocharging. The PL spectrum was measured on femtosecond time scales by using the transient grating technique described by Chen et al.51 A Ti:sapphire amplifier system (Spitfire Ace) operating at 3 kHz and generating 100 fs pulses was split into two parts. One part was converted to 400 nm using second harmonic generation and focused to a 70 μm spot on the sample. The PL was collimated using an off-axis parabolic mirror and refocused on a polished slice of fused silica. The second part of the fundamental 800 nm output was split using a 50:50 beam splitter creating two gate beams that were focused on the fused silica with a crossing angle of 5°. The instantaneous grating generated by the interfering gate beams created an instantaneous gate, which was used to temporally resolve the decay over a broad wavelength range. The scatter of the pump beam was suppressed by using a reflective geometry, and the pump was set to a magic angle relative to the PL collection. Data was averaged over 15000 shots for every time delay.

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(1) Pelant, I.; Valenta, J. Luminescence Spectroscopy of Semiconductors; Oxford University Press, 2012. (2) You, Y.; Zhang, X.-X.; Berkelbach, T. C.; Hybertsen, M. S.; Reichman, D. R.; Heinz, T. F. Observation of Biexcitons in Monolayer WSe2. Nat. Phys. 2015, 11, 477−481. (3) Cha, S.; Sung, J. H.; Sim, S.; Park, J.; Heo, H.; Jo, M.-h.; Choi, H. 1s-intraexcitonic Dynamics in Monolayer MoS2 probed by Ultrafast Mid-infrared Spectroscopy. Nat. Commun. 2016, 7, 1−7. (4) Sharma, A.; Yan, H.; Zhang, L.; Sun, X.; Liu, B.; Lu, Y. Highly Enhanced Many-body Interactions in Anisotropic 2D Semiconductors. Acc. Chem. Res. 2018, 51, 1164. (5) Wang, G.; Chernikov, A.; Glazov, M. M.; Heinz, T. F.; Marie, X.; Amand, T.; Urbaszek, B. Colloquium: Excitons in Atomically Thin Transition Metal Dichalcogenides. Rev. Mod. Phys. 2018, 90, 21001. (6) Weidman, M. C.; Seitz, M.; Stranks, S. D.; Tisdale, W. A. Highly Tunable colloidal Perovskite Nanoplatelets through Variable Cation, Metal, and Halide Composition. ACS Nano 2016, 10, 7830−7839. (7) Bohn, B. J.; Tong, Y.; Gramlich, M.; Lai, M. L.; Döblinger, M.; Wang, K.; Hoye, R. L. Z.; Muller-Buschbaum, P.; Stranks, S. D.; Urban, A. S.; Polavarapu, L.; Feldmann, J. Boosting Tunable Blue Luminescence of Halide Perovskite Nanoplatelets through Postsynthetic Surface Trap Repair. Nano Lett. 2018, 18, 5231−5238. (8) Ithurria, S.; Tessier, M. D.; Mahler, B.; Lobo, R. P. S. M.; Dubertret, B.; Efros, A. L. Colloidal Nanoplatelets with Twodimensional Electronic Structure. Nat. Mater. 2011, 10, 936−941. (9) Nasilowski, M.; Mahler, B.; Lhuillier, E.; Ithurria, S.; Dubertret, B. Two-dimensional Colloidal Nanocrystals. Chem. Rev. 2016, 116, 10934−10982. (10) Li, J.; Luo, L.; Huang, H.; Ma, C.; Ye, Z.; Zeng, J.; He, H. 2D Behaviors of Excitons in Cesium Lead Halide Perovskite Nanoplatelets. J. Phys. Chem. Lett. 2017, 8, 1161−1168. (11) Kumagai, M.; Takagahara, T. Excitonic and Nonlinear-Optical Properties of Dielectric Quantum-Well Structures. Phys. Rev. B: Condens. Matter Mater. Phys. 1989, 40, 12359. (12) Steinhoff, A.; Florian, M.; Singh, A.; Tran, K.; Kolarczik, M.; Helmrich, S.; Achtstein, A. W.; Woggon, U.; Owschimikow, N.; Jahnke, F.; Li, X. Biexciton Fine Structure in Monolayer Transition Metal Dichalcogenides. Nat. Phys. 2018, 14, 1199−1204. (13) Poellmann, C.; Steinleitner, P.; Leierseder, U.; Nagler, P.; Plechinger, G.; Porer, M.; Bratschitsch, R.; et al. Resonant Internal Quantum Transitions and Femtosecond Radiative Decay of Excitons in Monolayer WSe2. Nat. Mater. 2015, 14, 889−893. (14) Ye, Y.; Wong, Z. J.; Lu, X.; Ni, X.; Zhu, H.; Chen, X.; Wang, Y.; Zhang, X. Monolayer Excitonic Laser. Nat. Photonics 2015, 9, 733− 737. (15) Schneider, C.; Glazov, M. M.; Korn, T.; Höfling, S.; Urbaszek, B. Two-dimensional Semiconductors in the Regime of Strong LightMatter Coupling. Nat. Commun. 2018, 9, 1−9. (16) Singh, S.; Tomar, R.; ten Brinck, S.; De Roo, J.; Geiregat, P.; Martins, J. C.; Infante, I.; Hens, Z. Colloidal CdSe nanoplatelets, a Model for Surface Chemistry/opto-electronic Property Relations in Semiconductor Nanocrystals. J. Am. Chem. Soc. 2018, 140, 13292− 13300. (17) Pelton, M. Carrier Dynamics, Optical gain, and Lasing with Colloidal Quantum Wells. J. Phys. Chem. C 2018, 122, 10659−10674. (18) Guzelturk, B.; Pelton, M.; Olutas, M.; Demir, H. V. Giant Modal Gain Coefficients in Colloidal II-VI Nanoplatelets. Nano Lett. 2019, 19, 277−282. (19) She, C.; Fedin, I.; Dolzhnikov, D. S.; Demortière, A.; Schaller, R. D.; Pelton, M.; Talapin, D. V. Low-threshold Stimulated Emission using Colloidal Quantum Wells. Nano Lett. 2014, 14, 2772−7. (20) Li, Q.; Lian, T. A Model for Optical Gain in Colloidal Nanoplatelets. Chem. Sci. 2018, 9, 728−734. (21) Grim, J. Q.; Christodoulou, S.; Di Stasio, F.; Krahne, R.; Cingolani, R.; Manna, L.; Moreels, I. Continuous-wave Biexciton

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.9b01607.

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REFERENCES

Additional detailed information on the sizing of the QWs, their linear optical properties, the carrier recombination model, details on the (polarization resolved) spectroscopy, and an in-depth discussion of the gain model, also in comparison to other theoretical models (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Pieter Geiregat: 0000-0001-7217-8738 Shalini Singh: 0000-0001-8607-8383 Justin M. Hodgkiss: 0000-0002-9629-8213 Zeger Hens: 0000-0002-7041-3375 Author Contributions

P.G. set up the experiments, was involved in all of the femtosecond spectroscopy, developed the quantitative analysis of the gain spectra and the gain model, and wrote the manuscript. R.T. carried out the femtosecond transient absorption spectroscopy. S.S. supervised the synthesis work. K.C. was involved in the femtosecond transient PL spectroscopy, J.M.H. supervised transient PL spectroscopy and wrote the manuscript. Z.H. set up the research, developed the theoretical description and wrote the manuscript. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS Z.H. acknowledges support by the European Commission via the Marie-Sklodowska Curie action Phonsi (H2020-MSCAITN-642656), the Research Foundation Flanders (project 17006602), and Ghent University (GOA no. 01G01513). P.G. acknowledges the FWO-Vlaanderen for a fellowship and a 3643


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The Journal of Physical Chemistry Letters Lasing at Room Temperature using Solution-processed Quantum Wells. Nat. Nanotechnol. 2014, 9, 891−895. (22) She, C.; Fedin, I.; Dolzhnikov, D. S.; Dahlberg, P. D.; Engel, G. S.; Schaller, R. D.; Talapin, D. V. Red, Yellow, Green, and Blue Amplified Spontaneous Emission and Lasing using Colloidal CdSe Nanoplatelets. ACS Nano 2015, 9, 9475−9485. (23) Diroll, B. T.; Cho, W.; Coropceanu, I.; Harvey, S. M.; Brumberg, A.; Holtgrewe, N.; Crooker, S. A.; Wasielewski, M. R.; Prakapenka, V. B.; Talapin, D. V.; Schaller, R. D. Semiconductor Nanoplatelet Excimers. Nano Lett. 2018, 18, 6948−6953. (24) Bisschop, S.; Geiregat, P.; Aubert, T.; Hens, Z. The Impact of Core/shell Sizes on the Optical Gain Characteristics of CdSe/CdS Quantum Dots. ACS Nano 2018, 12, 9011−9021. (25) Pietryga, M.; Park, Y.-s.; Lim, J.; Fidler, A. F.; Bae, W. K.; Brovelli, S.; Klimov, V. I. Spectroscopic and device aspects of nanocrystal quantum dots. Chem. Rev. 2016, 116, 10513−10622. (26) Ye, Z.; Waldecker, L.; Ma, E. Y.; Rhodes, D.; Antony, A.; Kim, B.; Zhang, X. X.; Deng, M.; Jiang, Y.; Lu, Z.; Smirnov, D.; Watanabe, K.; Taniguchi, T.; Hone, J.; Heinz, T. F. Efficient Generation of Neutral and Charged Biexcitons in Encapsulated WSe2 Monolayers. Nat. Commun. 2018, 9, 6−11. (27) Kim, J. C.; Wake, D. R.; Wolfe, J. P. Thermodynamics of Biexcitons in a GaAs Quantum Well. Phys. Rev. B: Condens. Matter Mater. Phys. 1994, 50, 15099−15107. (28) Nurmikko, A. V. Optical Gain and Excitonic Processes in Widegap Semiconductor Quantum Wells. Phase Transitions 1999, 68, 95−149. (29) Ding, J.; Hagerott, M.; Ishihara, T.; Jeon, H.; Nurmikko, A. V. (Zn,Cd)Se/ZnSe Quantum-well Lasers: Excitonic Gain in an Inhomogeneously Broadened Quasi-two-dimensional System. Phys. Rev. B 1993, 47, 10528−10542. (30) Kozlov, V.; Kelkar, P.; Nurmikko, A. V.; et al. Excitonic Molecules and Stimulated Emission in a ZnSe Single Quantum Well. Phys. Rev. B 1996, 53, 10837−10840. (31) Kreller, F.; Lowisch, M.; Puls, J.; Henneberger, F. Role of Biexcitons in the Stimulated Emission of Wide-gap II-VI Quantum Wells. Phys. Rev. Lett. 1995, 75, 2420−2423. (32) Naeem, A.; Masia, F.; Christodoulou, S.; Moreels, I.; Borri, P.; Langbein, W. Giant Exciton Oscillator strength and Radiatively Limited Dephasing in Two-dimensional Platelets. Phys. Rev. B 2015, 91, 8−12. (33) Achtstein, A. W.; Schliwa, A.; Prudnikau, A.; Hardzei, M.; Artemyev, M. V.; Thomsen, C.; Woggon, U. Electronic Structure and Exciton-phonon Interaction in Two-dimensional Colloidal CdSe Nanosheets. Nano Lett. 2012, 12, 3151−3157. (34) Pelton, M.; Ithurria, S.; Schaller, R. D.; Dolzhnikov, D. S.; Talapin, D. V. Carrier Cooling in Colloidal Quantum Wells. Nano Lett. 2012, 12, 6158−63. (35) Tomar, R.; Kulkarni, A.; Chen, K.; Singh, S.; van Thourhout, D.; Hodgkiss, J. M.; Siebbeles, L. D.A.; Hens, Z.; Geiregat, P. Charge Carrier Cooling Bottleneck Opens Up Nonexcitonic Gain Mechanisms in Colloidal CdSe Quantum Wells. J. Phys. Chem. C 2019, 123, 9640−9650. (36) Kunneman, L. T.; Tessier, M. D.; Heuclin, H.; Dubertret, B.; Aulin, Y. V.; Grozema, F. C.; Schins, J. M.; Siebbeles, L. D. A. Bimolecular Auger Recombination of Electron-hole Pairs in Twodimensional CdSe and CdSe/CdZnS Core/shell Nanoplatelets. J. Phys. Chem. Lett. 2013, 4, 3574−3578. (37) Li, Q.; Lian, T. Area- and Thickness-Dependent Biexciton Auger Recombination in Colloidal CdSe Nanoplatelets: Breaking the “Universal Volume Scaling Law: Nano Lett. 2017, 17, 3152−3158. (38) Kleinman, D. A. Binding Energy of Biexcitons and Bound Excitons in Quantum Wells. Phys. Rev. B: Condens. Matter Mater. Phys. 1983, 28, 871−879. (39) Singh, J.; Birkedal, D.; Lyssenko, V. G.; Hvam, J. M. Binding Energy of Two-dimensional Biexcitons. Phys. Rev. B: Condens. Matter Mater. Phys. 1996, 53, 15909−15913.

(40) de Leon, S.; Laikhtman, B. The Spin Structure of Quasi twodimensional Biexcitons in Quantum Wells. Europhys. Lett. 2002, 59, 728−734. (41) Shiau, S. Y.; Combescot, M.; Chang, Y. C. Electronic Structure and Absorption Spectrum of Biexciton obtained by using Exciton Basis. Ann. Phys. 2013, 336, 309−330. (42) Geiregat, P.; Maes, J.; Chen, K.; Drijvers, E.; De Roo, J.; Hodgkiss, J. M.; Hens, Z. Using Bulk-like Nanocrystals to Probe Intrinsic Optical Gain Characteristics of Inorganic Lead Halide Perovskites. ACS Nano 2018, 12, 10178−10188. (43) Schmitt-Rink, S.; Ell, C.; Haug, H. Many-body Effects in the Absorption, Gain, and Luminescence Spectra of Semiconductor Quantum-well Structures. Phys. Rev. B: Condens. Matter Mater. Phys. 1986, 33, 1183−1189. (44) Ding, J.; Jeon, H.; Ishihara, T.; Hagerott, M.; Nurmikko, A. V.; Luo, H.; Samarth, N.; Furdyna, J. Excitonic Gain and Laser Emission in ZnSe-based Quantum Wells. Phys. Rev. Lett. 1992, 69, 1707−1710. (45) Gourley, P. L.; Wolfe, J. P. Thermodynamics of Excitonic Molecules in Silicon. Phys. Rev. B: Condens. Matter Mater. Phys. 1979, 20, 3319−3327. (46) Saito, H.; Göbel, E. Picosecond Spectroscopy of Highly Excited CdS. Phys. Rev. B: Condens. Matter Mater. Phys. 1985, 31, 2360−2369. (47) Kasprzak, J.; Richard, M.; Kundermann, S.; Baas, a.; Jeambrun, P.; Keeling, J. M. J.; Marchetti, F. M.; Szymańska, M. H.; André, R.; Staehli, J. L.; Savona, V.; Littlewood, P. B.; Deveaud, B.; Dang, L. S. Bose−Einstein Condensation of Exciton Polaritons. Nature 2006, 443, 409−414. (48) Byrnes, T.; Kim, N. Y.; Yamamoto, Y. Exciton-Polariton Condensates. Nat. Phys. 2014, 10, 803−813. (49) Sanvitto, D.; Kéna-cohen, S. The Road towards Polaritonic Devices. Nat. Mater. 2016, 15, 1061−1073. (50) Turner, D. B.; Nelson, K. A. Coherent Measurements of Highorder Electronic Correlations in Quantum Wells. Nature 2010, 466, 1089−1091. (51) Chen, K.; Gallaher, J. K.; Barker, A. J.; Hodgkiss, J. M. Transient Grating Photoluminescence Spectroscopy: An Ultrafast Method of Gating Broadband Spectra. J. Phys. Chem. Lett. 2014, 5, 1732−1737.

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Thermodynamic Equilibrium between Excitons and Excitonic

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