Stimulated Emission through an Electron-Hole Plasma in Colloidal CdSe Quantum Rings.

Colloidal CdSe quantum rings (QRs) are a recently developed class of nanomaterials with a unique topology. In nanocrystals with more common shapes, such as dots and platelets, the photophysics is consistently dominated by strongly bound electron-hole pairs, so-called excitons, regardless of the charge carrier density. Here, we show that charge carriers in QRs condense into a hot uncorrelated plasma state at high density. Through strong band gap renormalization, this plasma state is able to produce broadband and sizable optical gain. The gain is limited by a second-order, yet radiative, recombination process, and the buildup is counteracted by a charge-cooling bottleneck. Our results show that weakly confined QRs offer a unique system to study uncorrelated electron-hole dynamics in nanoscale materials.

In the past decade, Cd-based nanocrystals with variable degrees of confinement have been developed through colloidal synthesis procedures.[1,2] The ability to tune the nanocrystal shape has led to materials with remarkable optical properties. Nanoplatelets, for example, show anisotropic emission,[3] giant modal gain coefficients,[4,5] and record low gain thresholds[6] due to the presence of excitons, electron–hole pairs bound by strong Coulomb attractions, that are stable even at high densities.[7] However, all of these shapes are topologically equivalent with a genus of zero. An interesting new topology with a genus of one for Cd-based nanocrystals was put forward by Fedin et al., who synthesized toroidally shaped CdSe nanocrystals, also called quantum rings (QRs).[8] Starting from CdSe nanoplatelets, a thermochemical reconfiguration in the presence of selenium (Se) resulted in the formation of thin toroids.[8,9] The few optical studies on QRs have thus far been performed in the regime of a single electron–hole pair per ring, where likely excitons are the dominant charge species. Even in this regime, the QRs already display interesting properties such as broken rotational symmetry.[10] A single study on ultrafast time scales indicated strong coupling of single electron–hole pairs to surface defects and fast charge trapping.[11] To date, however, nothing is known on the nature of elementary excitations in QRs at increasing pair density. As a consequence, various interesting high-density effects remain unexplored, such as exciton-unbound charge carrier balances,[5,7,12] multicarrier recombination mechanisms,[13] carrier cooling[7,14,15] and potential stimulated emission.[4,16,17]

In this work, we combine femtosecond transient absorption and photoluminescence spectroscopy to study the photophysics of high-quality CdSe QRs over a full range of densities (i.e., from single up to 35 excitations per QR). First, we show that the used QR synthesis leads to limited ultrafast carrier trapping and pristine fast hole cooling at low carrier density, providing an intrinsic test bed for exploring QR properties. Next, we show that the dominant species at increasing carrier densities are unbound electron–hole pairs condensed into a plasma state which give rise to a sizable optical gain window. The latter is strongly red-shifted from the absorption spectrum due to band gap renormalization and remains broadband due to the absence of residual exciton absorption lines. In the QRs studied here, recombination at increasing carrier density is dominated by a cooling bottleneck at early times, followed by a fast yet radiative second-order recombination.

Figure a shows an electron microscopy image (HAADF-STEM) of the QRs which were synthesized using a modified procedure from Fedin et al.,[8] where the QRs are obtained starting from 4.5 monolayer (ML) CdSe nanoplatelets (Supporting Information).[9] As shown in the schematic of Figure a, the geometry of our QRs is best described as a single torus with outer dimensions L = 12.7 ± 6.0 nm and L = 10.3 ± 3.1 nm and an average in-plane width of w = 3.7 ± 0.9 nm. The thickness in the direction orthogonal to the ring plane amounts to L = 2.8 ± 0.6 nm, which equates to nearly 8.5 ML of CdSe.[1,18]

Figure 1

Overview of CdSe quantum rings. (a) HAADF-STEM imaging of the quantum rings (QRs) used in this work and a schematic depiction of a single QR with average outer in-plane dimensions of L = 12.7 nm and L = 10.3 nm and a thickness of L = 2.8 nm. (b) Numerical calculation of the distribution of the squared local field factor f2 in a CdSe QR for the external electric field (blue arrows) aligned along the in-plane x (left) and y (middle) directions and the out-of-plane z (right) direction. (c) Deconvolution of the (intrinsic) linear absorption spectrum μ using heavy-hole (HH, black), light-hole (LH, red), and spin–orbit split-off (SO, blue) exciton bands (solid), combined with free carrier contributions (dashed lines). A scheme of the ensuing optical transitions is shown in the inset. The binding energy ΔHH of the HH exciton is extracted from this fit as 107 ± 1 meV. (d) Transient absorption map ΔA after photoexcitation at 510 nm (2.4 eV), creating an average number of electron–hole pairs ⟨N⟩ = 13.8. The distinct optical transitions identified in Figure c are indicated by the vertical white dashed lines. The black contour line indicates the time–wavelength range where the bleach ΔA = A – A0 exceeds the linear absorption A0, thereby giving rise to a net optical gain (A < 0, g > 0, main text). (e) False color map of the PL under similar excitation conditions (⟨N⟩ = 18.5) as (d). The black trace indicates the steady-state photoluminescence spectrum which is centered at 627 nm.

Considering their complex shape, we first performed a numerical calculation to analyze the dielectric response of the QRs to an external electric field. For this, we use our method based on the quasi-static field approximation[19,20] (Supporting Information). In Figure b, we report the distribution of the local field factor, the ratio between the external and internal fields, inside the ring for the in-plane (x, y) and perpendicular (z) directions. Taking the integral of the local field distribution over the ring volume, we finally obtain that the linear intrinsic absorption coefficient μ at 309 nm amounts to 3.01 × 105 cm–1, which allows us to normalize (gain) spectra further on and calculate the number of absorbed photons per QR ⟨N⟩ accurately.[21]

In Figure c, we report the full absorption spectrum which resembles that of 2D semiconductors with a staircase profile toward higher energy.[1,18] The band gap absorbance consists of a broad peak at ∼1.9 eV,[17] associated with heavy hole (HH) and light hole (LH) transitions, and a second feature at higher energy which we assign to transitions from the split-off valence band (SO) to the conduction band. Next, we proceed with a quantitative decomposition in Figure c using a series of step functions, describing free carrier absorption from a 2D continuum of states (C, dashed lines) and the associated exciton absorption peaks (X, solid lines).[9] Notably, the exciton position energies, HH–LH splitting and the SO–CB transition extracted from the fit, line up with the values reported for weakly confined NPLs thicker than 8.5 ML (2.3 nm) (Supporting Information).[18,22] Finally, we extract the exciton binding energy ΔHH as 107 ± 1 meV, matching the trend for decreasing exciton binding energy for increasing thickness.[23] On the basis of both the model and geometrical description of the QRs, we can state that charges are only weakly confined in every direction for the QRs under study.

To study the charge carrier dynamics, we employ two ultrafast optical methods: transient absorption (TA) and femtosecond photoluminescence (fPL) spectroscopy. In both methods, a 110 fs pulse excites a colloidal dispersion of QRs. Next, a broadband scheme measures either the change in absorption ΔA = A – A0 due to the pump (TA) or the intensity of the luminescence (fPL).[24] We performed TA experiments pumping at 400, 510, and 625 nm adjusting ⟨N⟩ experimentally up to nearly 35. The fPL experiments were carried out by pumping at 515 nm, in a similar range of ⟨N⟩.

Figure d shows the result of a typical TA experiment (i.e., a time–wavelength map of the differential absorbance ΔA(λ, t)) for excitation at 510 nm using a photon flux that creates ⟨N⟩ = 13.8 electron–hole pairs per QR. Two distinct bleach bands (ΔA < 0) are observed around the SO/CB transition at 509 nm and around the band gap region at 620 nm. Pumping at 510 nm initially creates a hot hole in the SO valence band level and an electron in the lowest CB level (inset of Figure c). Focusing further on the band gap region at 620 nm, we observe a broadening at early times toward shorter wavelengths, which is absent at low pump power. At longer wavelengths (650–700 nm), a strong photoinduced absorption (PA, ΔA > 0, white area) gives way to a broad bleach region below the band gap after ca. 1 ps. When comparing this to the linear absorption A0, there is a region where ΔA exceeds A0, giving rise to a net negative absorbance A = ΔA + A0 < 0 or optical gain. The time-wavelength region where this occurs is strongly red-shifted from the band gap transition and is indicated by the solid black contour line in Figure d. Figure e shows the result of the fPL experiment, where the luminescence is collected over a similar time–wavelength scale and for a similar carrier density (⟨N⟩ = 18.5) as for TA. It reveals time scales similar to those of TA and a broadening toward longer and shorter wavelengths compared to the steady-state PL (black, Figure e).

To analyze the optical gain development in more detail, we will use the absorbance A(λ, t) = ΔA + A0, which we normalize to represent the intrinsic absorption μ(λ, t). First, we study the spectral dependence and threshold behavior by using spectral slices at a fixed delay of 3 ps. Figure a shows the result for increasing occupation ⟨N⟩. We observe a strong bleach at the band gap transition, eventually giving way to net optical gain μ < 0. The full collapse and even inversion of the lowest optical transition band upon strong photoexcitation are remarkable since they are not observed in the parent CdSe nanoplatelets.[5,7]Figure b shows the material gain g(λ) = −μ(λ) at 3 ps. The spectrum is initially confined to the region of 650 nm, where almost no linear absorption (dashed black line, Figure b) is present, only to blue shift toward 600 nm with a concomitant increase in magnitude, peaking at ca. 6000 cm–1. Having normalized to obtain the material gain, we can compare these numbers to other reports such as those for CdSe QDs (<3000 cm–1)[25] and CdSe nanoplatelets (>15.000 cm–1).[5,7]

Figure 2

Gain spectroscopy of quantum rings. (a) Slices of the nonlinear intrinsic absorption μ at 3 ps after photoexcitation with 510 nm. The carrier density is expressed as ⟨N⟩. Note the collapse of the band edge absorption feature at 625 nm (vertical arrow) and the occurrence of net optical gain μ < 0 beyond 600 nm. (b) Material gain g extracted from (a). The dashed line represents μ for the reference. (c) Material gain g (bottom) and gain lifetime τ (top) at 660 nm for increasing pair density and varying excitation wavelength: 400, 510, and 625 nm. The horizontal dashed gray line represents the value of μ at 660 nm. (d) Photoluminescence spectra for similar excitation conditions as in (a). Clear nonlinear increases on the low-energy (red shaded) and high-energy sides (blue shaded) are observed. (e) Gain models and an explanation of the photoinduced absorption in the exciton (top) and free charge carrier model (bottom). Dashed red bands indicate the red-shifted renormalized band gap Eg,r relative to the linear gap Eg,0 (black dashed bands). (f) Polarization-resolved pump–probe spectroscopy using circularly polarized pump (σ+) and probe (σ+(green dashed)/σ–(black)) sequences. Dashed vertical lines indicate the position of the ground-state exciton transition bleach (0 → X, red) and expected induced exciton-molecule transition (X → M, blue).

To evaluate the gain threshold, we plot in Figure c (bottom) the material gain at 660 nm for increasing electron–hole pair density ⟨N⟩, both for pumping resonantly with the HH/LH–CB (625 nm, red) and SO–CB (510 nm, green) transitions and for a scenario of hot excitation (400 nm, blue). For increasing density, we observe a crossover to net gain which occurs at ⟨N⟩ = 2.4, 3.5, 5.3 for 625, 510, and 400 nm pump wavelengths, respectively. The horizontal gray line in Figure c indicates the linear intrinsic absorption coefficient at 660 nm which is exceeded by a factor of 2, indicating strong spectral shifts.[25]

We finally proceed to study the gain lifetime, τ, defined as the time window for which g(t) remains positive. Figure c (top) shows that τ is fluence-dependent yet independent of the initial pump wavelength. The gain lifetime reaches 160 ps, on par with reports on core-only CdSe quantum wells[5] yet shorter than record lifetimes for core/shell architectures such as CdSe/CdS quantum dots and wells.[6,25,26] Remarkably, no saturation of the gain lifetime is observed, suggesting that τ could further increase at higher fluence. The question remains as to whether this lifetime is capped by radiative or nonradiative processes, a point we will discuss below. Figure d shows slices of the fPL experiments at 3 ps for increasing ⟨N⟩ over a range similar to that for the TA experiments. We observe a strong asymmetric broadening of the PL relative to the steady state (vertical gray line), showing a progressive increase in the signal at short wavelengths (blue) and a moderate increase with clear saturation at longer ones (red).

Both the TA and fPL experiments show remarkable blue and red shifts of the bleach and luminescence spectra. Combined with this, we observe a full inversion of the absorption spectrum into a continuous, broad gain spectrum which is not observed in highly excitonic 2D materials.[7] To understand these effects and their impact on optical gain, we should first consider how net stimulated emission could develop in weakly confined semiconductors. A first and distinctively exciton-mediated mechanism was recently unveiled for CdSe nanoplatelets.[5] In this model, the fusion of mobile and weakly confined 2D excitons to energetically favorable (biexcitonic) molecules gives rise to net stimulated emission along the molecule-to-exciton recombination pathway. A hallmark of this mechanism is the formation of molecules with well-defined total angular momentum.[5] Experimentally, a circularly polarized pump–probe sequence σ+σ+ should generate unbound molecules with total angular momentum F = 2, whereas σ+σ– is set to create a bound singlet molecular state (F = 0) by photon absorption from a single exciton state. Since possible molecular states are stabilized with a binding energy ΔM relative to the single exciton states, one expects an induced absorption at a probe photon energy of EX – ΔM for the σ+σ– sequence and the absence thereof in the σ+σ+ sequence. Figure f shows the result at 3 ps for such an experiment where the QRs were excited with polarization sequences as explained above. No difference is observed between the two pump–probe experiments, indicating that stable molecules cannot be formed in our QRs, thereby ruling out the excitonic gain mechanism.

A second mechanism for developing net gain is that found in many epitaxial quantum wells where charge carriers condense into an uncorrelated plasma-like state, leading to population inversion and net stimulated emission across the band gap (Figure e, bottom). The presence of a plasma state in 2D semiconductors normally leads to a red shift of the single-particle gap, called band gap renormalization (BGR).[27−29] The latter shifts the gain window toward lower energies, which is exactly what is observed in Figure b.

Further proof of a plasma state can be found in its peculiar dynamics at both early times and at later times. Before we continue to the high-density regime, we first assess the quality and intrinsic nature of the QRs through the single-charge pair dynamics.

Figure a shows the normalized TA traces probing the HH/LH–CB (dashed line) and SO–CB (solid line) transitions after excitation into the SO–CB (510 nm) manifold, hence creating a hot hole and a cold electron. Any discrepancy between both probe wavelengths would indicate hole dynamics, whereas common dynamics are to be linked to electrons.[30] At early times, we observe a common fast ingrowth scale indicative of fast occupation of the CB state by an electron. The slower, 300 ± 15 fs, ingrowth of HH/LH (dashed line) and decay of SO (solid line) indicate efficient hole cooling between SO and HH/LH (Supporting Information).

Figure 3

Dynamics of single electron–hole pairs in CdSe QRs. (a) Normalized ΔA traces probed at the HH/LH–CB transition (dashed line) and the SO/CB transition (solid line) after excitation at the SO/CB (510 nm) creating ⟨N⟩ = 0.5. A double-exponential fit (dotted red line) reveals fast single-hole cooling (300 fs) from the SO to the HH/LH level(s). (b) Normalized ΔA kinetics at longer time delays, with a log scale for the time axis, probed at the HH/LH–CB and SO–CB transitions indicating that after initial fast hole cooling a common electron trap depletes the conduction band on a 25 ps time scale (gray shaded area). However, the weight of this component is limited to 30% of the total decay.

Normalized after 2 ps, Figure b shows that all traces line up, indicating a loss of the common charge (i.e., electrons). As highlighted by the shaded region, we can identify two lifetimes of the signal decay: a fast but fluence-independent scale spanning a time window of ca. 25 ps, followed by a slower channel that accounts for >70% of the initial population. Notably, a similar set of observations was made by Xiao et al., who reported a substantially faster short lifetime (ca. 6 ps) and a much larger weight of the short-lived component.[11] The moderate trapping, both in relative weight and absolute time scale, in our QRs indicates their excellent surface passivation. The second regime, extending beyond 25 ps, shows a fluence-dependent lifetime, a point we will come back to further.

Next, we can proceed to understand what happens at increasing density. Looking back at Figures e and 2d, we observe strong luminescence blue-shifted from the steady-state PL spectrum in the first 10 ps time window, which indicates that charge carriers occupy a continuous band of high-lying energy levels.

It is well established that the slope of the luminescence at high photon energy can be described using a Boltzmann tail e–BTeff,[31−33] with Teff being the effective carrier temperature. Using this approach, Figure a shows a progressive increase of Teff with ⟨N⟩. In the same way, from the change in the fPL slope as a function of time, we can extract the cooling curves T(t) for increasing ⟨N⟩. As shown in Figure b, at low density, the carrier temperature drops monoexponentially on a subpicosecond time scale (τ0 = 270 ± 30 fs), matching the hole-cooling kinetics reported in Figure a. However, at increased pair densities, the temperature increases at early times and remains overheated until nearly 100 ps. A qualitative biexponential fit to the cooling curve at ⟨N⟩ = 1.9 (⟨N⟩ = 3.9) reveals an additional component, τ1 = 7 ± 1 ps (τ1 = 21 ± 2 ps) (Supporting Information). Charge cooling in polar semiconductors such as CdSe takes place by rapid optical phonon emission. However, it is exactly this rapid phonon emission which could overheat the phonon modes at high carrier density, leading to a lingering effective high carrier temperature Teff. To be more quantitative, we extract the average time required to emit a phonon τph from the slope of the cooling curves T(t)[34] (Supporting Information 3.10). As shown in Figure c, at low density τph is essentially constant at around 50 fs, matching a theoretical estimate for CdSe (Supporting Information 3.10). At higher electron–hole pair densities, τph markedly increases to nearly 10 ps. Similar observations were made on various polar semiconductors such as perovskites[35,36] and for various topologies of Cd-chalcogenides.[7,14,15,37] We note that the measured high carrier temperatures are incompatible with the thermal stability of excitons or biexcitons, again pointing to a plasma state.

Figure 4

Dynamics of electron–hole pairs at high density. (a) fPL spectra at 3 ps plotted on an energy scale for excitation at 515 nm (2.4 eV) at increasing pair density ⟨N⟩. Solid lines indicate the single-exponential Boltzmann fit used to extract the carrier temperature from the high-energy side of the fPL spectra. (b) Extracted effective carrier temperature from the slope of the spectra in (a) for ⟨N⟩ = 0.7, 1.9, 3.9. Biexponential fits (dashed lines) are discussed in the main text. (c) LO–phonon emission time extracted from the cooling curves in (b) as a function of the effective carrier temperature. The dashed black line represents the expected theoretical value for electron–LO phonon scattering. (d) Decay of ΔA integrated across the HH/LH – CB bleach fitted with a second-order recombination model (red lines) for ⟨N⟩ between 0.5 and 6.9. The inset represents the two-body recombination time constant .

After a few tens of picoseconds, the charge cooling is complete and the limited linear electron trapping (25 ps) is finished. Beyond this time frame, we observe that the recombination is dependent on the pump fluence (Figure d). In general, carrier recombination can be modeled using a mixed first- and second-order process:

However, in the range beyond 25 ps, the (electron) trapping processes summarized by k1 can be neglected. As shown in Figure d, a fit with k2 as a global parameter matches the ΔA traces for ⟨N⟩ between 0.5 and 6.9. The fit yields a second-order recombination rate of k2 = (7.7 ± 0.1) × 10–3 cm2/s, which is in line with various reports.[38−41] However, it exceeds those values found for thinner 4.5 ML CdSe NPLs by an order of magnitude (k2 = 0.13 × 10–3, 0.97 × 10–3 cm2/s),[7,42] most likely because the process considered here for unbound charge carriers entices a radiative electron–hole recombination process, which contrasts with Auger-type exciton–exciton annihilation. Our analysis implies that the limited gain lifetime is hence not due to nonradiative losses but is due to fast radiative recombination, an important difference with nanoplatelets or quantum dots and a feat also observed in weakly confined perovskites.[35,43] From k2, we can calculate a density-dependent two-body recombination time (Figure d) which varies between 0.1 and 1 ns.

We obtained a set of intrinsic observations on QRs which point toward a situation where not excitons but a hot plasma state is present, giving rise to peculiar photophysics that is unexpected for the case of a confined 2D material. Both the full exciton saturation and inversion, the absence of polarization-resolved transitions, and the high carrier temperatures make exciton-based gain unlikely. Clearly, the reduced confinement in the out-of-plane direction compared to thin nanoplatelets, combined with strong carrier–carrier interactions evidenced by strong band gap renormalization and comparatively fast radiative bimolecular recombination, strongly reduces the excitonic nature of the fundamental excitations in QRs. We did not observe any correlation between the QR’s unique g = 1 topology and the ultrafast phenomena discussed here which could be due to limited rotational symmetry, as observed similarly by Hartmann et al.,[10] or carrier localization at room temperature.[44] Overall, CdSe QRs at room temperature behave more as classical quantum wells, providing a unique platform for studying unbound charge carriers in colloidal materials.