A Thermodynamic Equilibrium between Excitons and Excitonic

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Spectroscopy and Photochemistry; General Theory

A 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 J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.9b01607 • Publication Date (Web): 12 Jun 2019 Downloaded from http://pubs.acs.org on June 12, 2019

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The Journal of Physical Chemistry Letters

A Thermodynamic Equilibrium between Excitons and Excitonic Molecules Dictates Optical Gain in Colloidal CdSe Quantum Wells Pieter Geiregat,∗,†,‡ Renu Tomar,†,‡ Kai Chen,¶,§,k Shalini Singh,†,‡ Justin M. Hodgkiss,¶,§,k and Zeger Hens†,‡ †Physics and Chemistry of Nanostructures, Ghent University, Ghent, 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 kThe Dodd-Walls Centre for Photonic and Quantum Technologies, Dunedin, 9054, New Zealand E-mail: [email protected]

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ACS Paragon Plus Environment


Abstract We show that optical gain in 2D CdSe colloidal quantum wells (CQWs) shows little saturation and co-exists 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 photo-excited 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 the remarkable gain characteristics of CQWs using only experimentally determined parameters, an advance that highlights a fundamental difference between multi-excitons in CQWs and CQDs. While strong confinement prohibits the dissociation of multi-excitons 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.

Graphical TOC Entry e h

e

e

+

ü SurfaceΔ Density M = 45 meV ü BindingT= Energy 400 K ü Temperature

h

+

e 10

=9

h

=3 h

,

e

10

=9

e

=3

20.000 cm-1

N=15

h e

N=15

1 R=0.34

0 1

1

2

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6 8

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1 R=0.34

2

0


1

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h e

“Molecule”

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ΔM = 45 meV T= 400 K

h

h

⇌

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Excitons (X) are composite quasi-particles consisting of an electron and a hole held together by Coulomb attraction, see Figure 1a. Electronic transitions involving free excitons exhibit a strong light-matter interaction, yet the small exciton binding energy ∆X of only 1-10 meV restricts excitonics in bulk semiconductors to cryogenic temperatures. In twodimensional (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 monoand multi-layers 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 ∆X 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 low-screening 2D materials abound, 2,12,13 and the numerous reports on optical gain and lasing attest to the promise of using these materials in opto-electronics. 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 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 twoexciton states, 20–22 or excimers formed by charge transfer between adjacent nanoplatelets. 23 Even though a consensus seems to be that multiple excitons are required per CQW, no quantitative and/or full consistent models exist to date. 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

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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 multi-exciton states in QDs and QWs. Since the QD volume is smaller than the volume dictated by the exciton Bohr radius, a bi-exciton in a QD is simply the combination of two electron-hole pairs, regardless of the attractive or repulsive Coulomb interaction between both pairs. 25 The large lateral area of a QW, on the other hand, makes that two electron-hole pairs can either exist as two separate, dissociated and mobile excitons or as a single, associated quasi-particle consisting of two bound excitons (see Figure 1a). Clearly, such a quasi particle – henceforth called the excitonic molecule (M) – will only form in 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 a

CQW area >> π a B

EM

e

h

h

e

2

bE

KM

?

e

EX e h

KX

|ψM>

KX

X→M

EM,0=EG-ΔX-ΔM

0→X

EX,0=EG-ΔX

|ψX>

EX

h

0

|ψ0> -K

0

K

Figure 1: (a) Schematic representation of (red, X) excitons as electron-hole pairs bound bound by Coulomb interaction with binding energy ∆X and (blue, M) an excitonic molecule as an adduct of two excitons with a binding energy ∆M . 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 πa2B much smaller than the total CQW area, hence making the fusion of 2X into M not obvious. (b) Energy dispersion of 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. 4


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of molecules, 26–29 and the formation of these bound quasi-particles at room temperature is considered non-trivial, 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 in-plane wavevector K, see figure 1b, also the photon-assisted internal transitions between X and M are non-trivial and different from the 0D case where translational momentum is not necessarily conserved. Here, we show that 2D excitons in CdSe CQWs can associate to form stable excitonic molecules with a binding energy ∆M 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 equilbrium, 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 recombination presents a conceptual leap forward in terms of the mechanistic understanding of multi-exciton physics in colloidal 2D materials. For this study, we synthesized two batches of 4.5 monolayer (1.37 nm) CdSe CQWs, large CQWs measuring 34×9.6nm2 , and small CQWs measuring 15.4×5.7nm2 using the procedure first proposed by Ithurria et al. (see Supporting Information 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 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

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ΔX 190 meV

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0.5 0.0

10 2 1 0 -1

IPL=0.01

480 520 560 wavelength (nm)

600

Figure 2: (a) Brightfield TEM image of the large CdSe colloidal quantum wells. (b) Bottom. Spectrum of (red line) intrinsic absorption coefficient and (filled red) photoluminescence of the large CdSe CQWs. The position of the heavy hole (HH) and light hole (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 ∆X = 193 ± 5 meV is indicated. (c) 2D delay time/wavelength map of ∆µi after femtosecond photo-excitation with 400 nm (3.1 eV) pulses with an energy of 45µJ/cm2 . The contours (dashed) PA and (dotted) gain indicate areas where a photo-induced 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 photo-excitation with similar pump conditions as in (c). The black line represents the steady-state PL spectrum. 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 only shows 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 Supporting Information S2). As outlined in Supporting Information S3, this spectrum can be decomposed into separate contributions of 6


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the exciton and free carrier absorption from the HH and LH states, respectively. In line with previous literature reports, this yields a heavy hole exciton binding energy ∆X = 193±5meV, see Figure 2b. 32,33 Figure 2c represents the delay time/wavelength map of the transient absorption coefficient ∆µi of the large CQWs after photo-excitation using 400 nm, 45 µJ/cm2 femtosecond pulses, which create on average hN i = 7.5 excitations per CQW (see Supporting Information S4). In line with previous reports, we found that ∆µi 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 second order/first order process (see Supporting Information S5). 36,37 More intriguing elements of the ∆µi 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 photo-induced absorption, which peaks 45 meV to the red of the HH exciton line. With increasing delay, this photo-induced 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 photo-induced 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 ∆M that should be 0.228 × ∆X 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. Since 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 for-

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σ σ

50

0 -50 E-EX (meV)

X→M

µ i,0

0

-100

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=1.84 10 cm

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Figure 3: (a) Diagram indicating the optical transitions between the different spin-polarized states of the exciton X and the molecule M. Dashed grey lines indicate the dark J = ±2 exciton states. (b) ∆A map recorded on the large CQWs after pumping at 510 nm creating hN i = 8 excitations per platelet using (bottom) co-circular σ + σ + and (top) counter-circular σ + σ − pump-probe polarization. (c) ∆A 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 shown. (d) Integrated intrinsic absorption coefficient µX→M (E) versus excitation number. The linear scaling yields i the absorption coefficient µX→M = 1.84 × 106 cm−1 . i,0 mation 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 ∆A maps measured using σ + and σ − probe light, after resonant excitation with 510 nm σ + pump light, see Methods and Supporting Information 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 only leads to a gradual 8


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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 spin-polarized exciton state across all available exciton states, see Supporting Information S6. As shown in Figure 3c, we obtain the absorbance spectrum of the X → M transition by a deconvolution of the counter-polarized 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 ∆M . Additionally, this spectrum can be understood by considering the dispersion of the center-of-mass motion of the exciton and the molecule, see Figure 1b. As outlined in Supporting Information S7, the absorption coefficient µX→M (E) can be written as the product of an intrinsic absorption coefficient µX→M i i,0 of the transition and the probability f that the concomitant center-of-mass state is occupied. Referring to Figure 1b and Supporting Information S7, we thus have:

µX→M (E) = µX→M f (2(EM,0 − E)) = µX→M i i,0 i,0

−E π¯ h2 hN i −2 EM,0 kB T e mX kB T S

(1)

Here, we replaced f by a Boltzmann distribution, denoted the exciton mass as mX and use 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 i,0 best obtained from the energy integrated spectrum. As indicated in Figure 3d, this integral scales proportionally to hN i with a slope yielding µX→M = 1.84 × 106 cm−1 . i,0 Figure 4a represents a map of the material gain gi = −(µi + ∆µi ) 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 hN i = 15 excitations per platelet. In

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10

5

=4.1 =1.2

0

0 510

540 wavelength (nm)

570

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Figure 4: (a) Time-wavelength map of the material gain gi after photo-excitation at 400 nm with a pulse energy of 90 µJ/cm2 ; a pulse creating on average hN i = 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 hN i 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 photo-excitation as a function of the average exciton density hni for large and small CQWs. agreement with the ∆µi 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 different initial excitation numbers hN i. It can be seen that the onset of net optical gain at hN i ≈ 5 concurs with the appearance of the redshifted PL band that was already highlighted in the PL map shown in Figure 2d. As outlined in Supporting Information 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 10


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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 hNth i = 4.1 for the large CQWs, whereas the small CQWs feature a threshold at hNth i = 1.2 (see Figure 4c and Supporting Information S9). Normalizing these thresholds by the CQW area, we find that both numbers correspond to the same threshold excitation density hnth i of 0.013 nm−2 , which is attained at a threshold fluence of 25 µJ/cm2 . Further, using hni 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 hni. Moreover, even at excitation densities that are a 10-fold of hnth i, no saturation of gi occurs. As outlined in Supporting Information 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 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, spectral 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 coefficient µX→M (E) of a mixed population of free excitons and molecules amounts to the sum of i 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 i

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0 0

0.0

40 80 0.1 1 t (ns) t (ps)

0.1 0.2 EM,0 - E (eV)

0.3

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 hNX i and molecules hNM i and their ratio R = hNM i/hNX i, as function of total excitation number hN i. An example case of hN i = 15 is highlighted. (c) Excitation number as extracted from time-resolved photoluminescence. Time delays of 20, 40 and 60 ps correspond to hN i = 36, 22 and 15 respectively. The inset shows the temperature of the exciton gas as extracted from the PL spectrum, see Supporting Information S11. (d) Representation of (cross markers) experimental and (solid lines) simulated material gain spectrum, showing an excellent correspondence. The experimental spectra corresponds to the delays indicated in (c), and the corresponding hN i and T where used in the simulation. (see Supporting Information S11):

µX→M (E) i

=

µX→M i,0

−E π¯ h2 hNX i − EM,0 e kB T 4mX kB T S

E −E − M,0 e kB T − 2R

(2)

Here, E is the photon energy and R denotes the ratio hNM i/hNX i between the molecule and the exciton number. Eq 2 indicates that the absorption coefficient of a given mixture of

12


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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 the 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 , hN i and i,0 R must be known. While pump-probe spectroscopy already yielded EM,0 and µX→M , we rely i,0 in what follows on an association/dissociation equilibrium between excitons and molecules to obtain R, see Figure 5b and Supporting Information S10. 27 This is a key assumption which differs fundamentally from state-filling models put forward to describe optical gain by CQWs. 20 Indeed, state-filling identifies a state of 2 excitons in a CQW with a biexciton, a state of 3 excitons with a tri-exciton, and so on. A thermodynamic equilibrium on the other hand, allows for the hN i = 2 exciton state to exist as 2 separate excitons or a single molecule, whereas the hN i = 3 state can correspond to 3 separate excitons, an exciton and a molecule or, rather unlikely, a tri-exciton molecule. Restricting the bound states to excitons and molecules, the average number of both quasi-particles per CQW follows from a dissociation equilibrium: 27 2X * )M

(3)

This equilibrium requires that the number of excitons hNX i and molecules hNX i are related by the equilibrium equation (K: equilibrium constant): hNM i =K hNX i2

(4)

Denoting the initial number of excitations per CQW as hN i = hNX i + 2hNM i, the average number of excitons and molecules per CQW can then be expressed as: p

8KhN i + 1 − 1 4Kp 4KhN i + 1 − 8KhN i + 1 hNM i = 8K hNX i =

13


(5) (6)


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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 Supporting Information S10): 27,45,46

K=

∆ 4π¯h2 S − M e kB T 2 gX mX kB T

(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 hNX i and hNM i with increasing excitation number hN i, calculated according to the approach outlined above and taking the degeneracy of the heavy-hole exciton gX = 4 and the molecule binding energy ∆M = 45 meV (see Supporting Information S10). For an electronic temperature of 400 K, Figure 5b shows that large CQWs will hold 1 molecule when hN i = 7, an occupation only slightly above the gain threshold for such CQWs. Moreover, since 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 hNX i/S and hNM i/S as a function of the excitation density hN i/S. Hence, the Eq 2 yields a net absorption or gain coefficient that is a function of the excitation density hN i/S, rather than the excitation number hN i, 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. Since Eq 2 only applies to 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 hN i and the temperature of the exciton-molecule gas, see Figure 5c and Supporting Information S11. Figure 5d displays three experimental gain spectra (markers) – taken at different pumpprobe delay times after a pump pulse creating hN i = 55 excitations – together with the gain spectrum as predicted by Eq 2 (solid lines). As can be seen, the assumption underlying

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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 multi-excitons are involved in the stimulated emission process, the current stateof-the-art provided no such insights in how to model the relative populations of excitons and multi-excitons, 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 derives from the fact in 0D QDs, the biexciton cannot dissociate to form two separate excitons. Since the QD volume is smaller than the exciton Bohr volume, two excitons in a 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. This makes that the multiple electron-hole pairs created by photo-excitation 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 Supporting Information S12, this fundamental difference between excitons in QDs and CQWs makes that optical gain and exciton absorption can co-exist 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 well-documented for QDs, the relation between exciton absorption and optical gain in

15



CQWs is different. The TA spectra shown here, see also Supporting Information S12, and in literature highlight that optical gain and exciton absorbance effectively co-exist over a broad range of excitation densities in CdSe CQWs. In the case of the large CQWs, for example, optical gain first occurs when hN i ≈ 4 while the exciton absorbance persists up to at least hN is ≈ 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 a thermodynamic formation of bound multi-exciton 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 of the X → M transition, which we estimate for CdSe CDQs at 1.8 106 cm−1 , coefficient µX→M i,0 and the probability that molecules occupy a given center-of-mass translational state. Since 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 high power, 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 only by the two-body recombination time, which can amount to several nanoseconds in CdSe CQWs, see Supporting Information S5. In addition, the high stability of excitonic molecules suggests that these quasi-particles will be

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the dominant species at cryogenic temperatures. This can significantly modify the process of exciton condensation since the inter-particle 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 phenomena 49 and higher order complexes such as tri-excitons without the need for cryogenic temperatures or complex coherent multi-photon interactions. 50 In summary, this study builds on a series of findings, which include the 20-year old observation of excitonic molecules and stimulated emission through the M → X transition for quantum wells at cryogenic temperatures, 30 and the recent demonstration that in monolayer transition metal dichalcogenides, 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 opto-electronics at room temperature. In this respect, colloidal quantum wells have multiple advantages. In particular the atomically precise and controllable synthesis, solution processability, the strong suppression of nonradiative exciton recombination thanks to a nearly perfect surface termination, 16 and the ideal fit between transparent nanocolloids and quantitative femtosecond spectroscopy stand out. In combination with the versatile device integration, colloidal quantum wells therefore offer a most feasible pathway to investigate, develop and implement room temperature excitonics.

Methods Synthesis of colloidal CdSe quantum wells. We synthesized the batches of 4.5 monolayer CdSe CQWs or colloidal quantum wells (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) 17



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. Afterwards, the reaction mixture was heated to 240◦ C for 10 minutes 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 minute 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 densities of 0.1 at the first heavy-hole exciton transition across a 1 mm cuvette.

Transmission Electron Microscopy. Brightfield transmission electron microscopy 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 PerkinElmer Lambda 950 UV-Vis-NIR spectrophotometer.

Steady-State Photoluminescence Spectroscopy. The quantum yield of the photoluminescence 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. For measuring the transient absorbance, dispersions of CdSe CQWs in hexane were photo-excited using 110 femtosecond (fs) pulses with varying wavelengths. These pump pulses were created from the amplified 800 nm fundamental of a Titanium-Sapphire laser (Spitfire Ace, Spectra Physics, 1 kHZ) through non-linear frequency mixing in an optical parametric amplifier (TOPAS, Light Conversion). The broad-

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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 maximum delay of 3 ns (Newport TAS). Noise levels of 0.1 mOD (RMS) are achieved by averaging over 10000 shots. The probe spectrum covers the VIS-NIR window from 350 nm 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 photo-charging 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 photoluminescence quantum yield when exposing samples to ambient conditions and/or the laser pulses. To obtain circular polarization for pump and probe, a combination of linear polarizers and (broadband) quarter-wave plates with high polarization contrast was used.

Ultrafast Luminescence Spectroscopy. For femtosecond photoluminescence (PL) spectroscopy, dispersions of CdSe CQWs were loaded in a cuvette with a 1 mm optical path length and an absorbance of 0.2 at the heavy-hole absorption peak to avoid strong re-absorption. Samples were translated along 1 axis to avoid photo-charging. The PL spectrum was measured on femtosecond timescales 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 photoluminescence is 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 are focused on the fused silica with a crossing angle of 5 degrees. The instantaneous grating generated by the interfering gate beams create an instantaneous gate which is 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

19



pump was set a magic angle relative to the PL collection. Data is averaged over 15000 shots for every time delay.

Acknowledgement ZH acknowledges support by the European Commission via the Marie-Sklodowska Curie action Phonsi (H2020-MSCA-ITN-642656), the Research Foundation Flanders (project 17006602) and Ghent University (GOA no. 01G01513). PG acknowledges the FWO-Vlaanderen for a fellowship and a travel grant. JMH and KC acknowledge support from the Marsden Fund.

Author Contributions PG set up the experiments, was involved in all the femtosecond spectroscopy, developed the quantitative analysis of the gain spectra and the gain model, and wrote the manuscript. RT carried out the femtosecond transient absorption spectroscopy. SS supervised the synthesis work. KC was involved in the femtosecond transient photoluminescence spectroscopy, JMC supervised transient photoluminescence spectroscopy and wrote the manuscript. ZH set up the research, developed the theoretical description and wrote the manuscript.

Supporting Information Available The Supporting Information contains more 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.

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A Thermodynamic Equilibrium between Excitons and Excitonic

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