Enhancement of the Biexciton Binding Energy in Laterally Confined CdSe Nanoplatelets

Abstract

Optical amplification in CdSe nanoplatelets (NPLs) has been linked to biexcitons with a large binding energy Δ_BX, preventing dissociation at room temperature. While the exciton binding energy Δ_X has been studied extensively, Δ_BX in colloidal NPLs is typically inferred using the 2D Haynes rule, Δ_BX = 0.228·Δ_X. Systematic studies of Δ_BX in CdSe NPLs with varying thicknesses and lateral dimensions are still lacking. Here, we used transient absorption spectroscopy to investigate Δ_BX, through photoinduced biexciton absorption. Δ_BX ranges between 40 and 55 meV and increases by 30% when reducing the width from 9 to 3 nm, while overall 3.5 ML NPLs have only 10% higher Δ_BX compared to 4.5 ML NPLs. Fitting the linear absorbance spectra, we extracted Δ_X and demonstrate that the Δ_BX/Δ_X ratio is substantially lower than 0.228. Results confirm that biexcitons in CdSe NPLs are stable at room temperature and have an increased Δ_BX in laterally confined NPLs.

Solution-processed nanoplatelets (NPLs) are promising materials for light emission and amplification due to their narrow and tunable photoluminescence, − high oscillator strength of the optical transitions, − large exciton − and biexciton ,,,− binding energies, and fast radiative decay lifetimes. ,− These desirable features have led to the demonstration of efficient light-emitting diodes (LEDs), − as well as amplified spontaneous emission (ASE) ,− and lasing ,− at room temperature. The low optical gain thresholds ,,, and high material and modal gain coefficients ,,, have been linked to a stable biexciton population, even at high carrier density. ,, This distinctive feature of colloidal two-dimensional materials arises from the combination of quantum and dielectric confinement of charge carriers, which enhances the electron–hole Coulomb interactions in these systems compared to bulk materials and epitaxial quantum wells, − leading to exciton binding energies largely exceeding thermal energy at room temperature. −

Excitons are quasiparticles formed by an electron and a hole bound by Coulomb interactions. In bulk semiconductors, Δ_X typically amounts to 1–15 meV. Confining excitons in one or more directions can increase Δ_X. Through a variational calculation, Δ_X can be written as

$${\mathrm{\Delta }}_{\mathrm{X}}=\left(\frac{\mu }{m_e{\epsilon }^2}\right){\left(\frac{2}{\alpha -1}\right)}^2{E}_H$$ 1

Here, μ is the reduced exciton mass (expressed in units of electron mass m _e), ε is the relative dielectric constant, α is a dimensional parameter that varies from 3 for bulk, over 2 for an infinitely thin quantum well, to 1 for a quantum wire, and E _H is the hydrogen binding energy of 13.6 eV. Reducing α from 3 to 2, 2D semiconductors already allow for a 4-fold increase of Δ_X. Note that eq is derived starting from a Hamiltonian, which is fractal for the kinetic energy but not for the Coulomb potential, which maintains a 1/r dependence. Because of this, the 1D limit leads to an infinite binding energy, which is not realistic. However, the equation does highlight that both transversal and lateral confinement increases Δ_X.

Two excitons can interact and form a biexciton. Typically, the energy that keeps the two excitons together (biexciton binding energy, Δ_BX) is proportional to Δ_X, which is known as the Haynes rule. Early studies on epitaxial GaAs quantum wells, based on effective mass calculations, suggested that Δ_BX/Δ_X = 0.1–0.15, depending on the well thickness. However, photoluminescence experiments yielded a constant Haynes factor Δ_BX/Δ_X ≈ 0.2, regardless of the thickness. The latter result was interpreted by Singh and co-workers as the limit of a squared biexciton in a strictly 2D system, where they found Δ_BX/Δ_X = 0.228.

Haynes’ rule has been previously applied to infer Δ_BX in colloidal 2D materials ,,, and transition-metal dichalcogenide nanosheets; a Haynes factor of 0.228 has also been assumed in CdSe , and CsPbBr₃ , NPLs, yet this has not been verified experimentally. In addition, epitaxial wells and NPLs present relevant differences. In contrast to epitaxial quantum wells, which are typically embedded in a cladding layer of similar dielectric constant, colloidal 2D NPLs are capped with an organic ligand shell and typically suspended in a low-dielectric solvent. In addition, the thickness is an order of magnitude smaller than in epitaxial wells (ca. 1 nm vs ca. 10 nm). Both dielectric and spatial confinement increase the electron–hole Coulomb interaction ,,, and thus Δ_X and Δ_BX. In the case of CdSe (ε = 6–10) suspended in an organic solvent (ε = 2) it can yield values on the order of 100–450 meV for Δ_X and 10–55 meV for Δ_BX (Figure ), which are much larger than Δ_X ≈ 5 meV and Δ_BX ≈ 1 meV obtained in GaAs quantum wells. This provides a unique testbed to probe the 2D Haynes rule using colloidal NPLs. From a theoretical perspective, Climente et al. have studied the Δ_BX/Δ_X ratio in CdSe and halide perovskite NPLs. The calculations highlighted that biexcitons in NPLs depart from an ideal flat square configuration. This results in Δ_BX/Δ_X < 0.2, with the actual ratio showing minor dependence on the NPL thickness, but a sizable one on the width dependence. By contrast, Δ_BX increases with both smaller thickness and width. The above controversy underlines the importance of determining accurate experimental values for both Δ_X and Δ_BX, if one wishes to employ 2D colloidal NPLs in efficient optical amplifiers and lasers that rely on stable biexciton populations ,− or as single-photon emitters with high purity, − which would benefit from a large exciton–biexciton energy splitting to spectrally filter out biexciton emission.

1.

(a) Exciton binding energies reported in the literature. Full circles correspond to values recorded using various spectroscopic techniques, − including current work (black). Open symbols show calculated values − using either the high frequency (ε = 6) or static (ε = 10) CdSe dielectric constant. (b) Biexciton binding energies reported in the literature (closed squares ,,− ) and in the current work (black), together with calculated values (open circles).

Δ_X in CdSe NPLs has been quantified using various spectroscopic methods and modeling tools, − as well tight-binding, variational quantum Monte Carlo (VQMC) simulations, and combined density functional theory/k·p calculations (Figure a). Despite significant scattering of the data, a general increase in Δ_X with decreasing NPL thickness is observed. The same trend has been reported for CsPbBr₃ NPLs. In addition to this, VQMC simulations predict an increase of Δ_X with decreasing NPL width, which could explain part of the scattering observed for a single thickness. Data on Δ_BX are scarcer and show a similar issue with scattering (Figure b).

We therefore systematically investigated the thickness and lateral dependence of Δ_X and Δ_BX in CdSe NPLs. To this aim, we analyzed linear and transient absorption spectra of a set of 3.5 and 4.5 ML CdSe NPLs with various lateral dimensions and carried out VQMC simulations in the same thickness and width range. We found that Δ_BX ranges between 40 and 55 meV, and the width reduction yields the most important variation of Δ_BX. We correlated these data with Δ_X derived from absorption spectra, using a quantum well fitting model with restricted fitting parameters derived from the effective masses of the electron (E) and heavy (HH), light (LH), and split-off (SO) holes. Finally, we obtained that the Haynes factor Δ_BX/Δ_X falls significantly below 0.228 for large NPLs (Δ_BX/Δ_X ≈ 0.15), yet it increases in laterally confined NPLs.

We carried out our investigation on a set of 3.5 and 4.5 ML of CdSe NPLs with various lateral dimensions. The NPLs were synthesized following established procedures, , tuning the width of the particles between 3 and 10 nm. Figure a,b shows representative transmission electron microscopy (TEM) images of 3.5 ML NPLs with lateral dimensions of 3.7 × 25.6 nm and 4.5 ML NPLs of 4.3 × 28.8 nm, respectively. Figure c displays the absorbance spectra of the investigated samples, showing the typical HH–E transitions at ∼2.4 eV and ∼2.7 eV for 4.5 and 3.5 ML, respectively. A slight variation of the HH–E transition within the set of samples with the same thickness can be observed, which can be ascribed to lateral confinement. , As highlighted in Figure d, the variation matches the empirical relation (solid lines) already reported for CdSe NPLs (d: NPL thickness, w: NPL width, in nm): ,

$$E_{HH}=1.49+\frac{1.27}{d}+\frac{0.40}{w^2}$$ 2

2.

(a, b) Representative TEM images of 3.5 ML (a) and 4.5 ML (b) NPLs, with lateral dimensions of 3.7 × 25.6 nm and 4.3 × 28.8 nm, respectively. The scale bar equals 50 nm. (c) Absorbance spectra of 3.5 ML (blue) and 4.5 ML (green) CdSe NPLs with various lateral dimensions. The spectra are normalized to the HH–E intensity and shifted for clarity. (d) Energy of the HH–E peak extracted from the absorption spectra in (c) plotted against the NPL width, as extracted from TEM images. (e, f) Representative absorbance spectra and fits for 3.5 ML (e) and 4.5 ML (f) CdSe NPLs, together with individual contributions of HH–E, LH–E, and, in the case of 3.5 ML NPLs, SO–E exciton and free-carrier transitions. (g, h) Resulting exciton binding energy Δ_X for 3.5 ML (g) and 4.5 ML (h) NPLs as a function of width (closed circles), together with calculated values using a dielectric constant of 10 (upward triangles). Data are fitted with an exponential function; see also SI, Figure S2.

We first extracted Δ_X by fitting the absorbance spectra, using the approach introduced for colloidal nanoplatelets by Naeem et al. Close to the band gap, the NPL absorbance spectrum can be decomposed in a series of exciton absorption lines and free-carrier absorption steps, originating from the HH–E, LH–E, and, in the case of 3.5 ML NPLs and laterally confined 4.5 ML NPLs (SI, Figure S1), SO–E transitions. The associated equations are reported in Supporting Information, Section S3. The procedure is straightforward, yet has led to relatively low and disparate values of Δ_X, ranging between 130 and 180 meV. ,, We therefore restricted fitting parameters to improve consistency. First, the ratio of the HH–E and LH–E exciton binding energies should equal the ratio of their in-plane reduced exciton masses, μ _LH/μ _HH (see also eq ). Second, the strength (amplitude) of the free-carrier absorption depends on the transition dipole moment (equal for HH–E and LH–E transitions), multiplied by the joint density of states ρ _2,D = μ/(ℏ ²π). Hence, the amplitude ratio is also equal to the ratio of the HH–E and LH–E reduced exciton masses. Considering these arguments, and using reduced exciton masses derived from the effective electron and hole masses reported by Benchamekh et al., μ _X,LH/μ _X,HH amounts to 1.04 and 1.02 for 4.5 and 3.5 ML CdSe NPLs, respectively. These fixed ratios also apply to Δ_X,LH/Δ_X,HH and A _C,LH/A _C,HH and are used in the fits to the absorbance spectrum. Figure e,f shows the overall result and individual HH–E, LH–E, and SO–E contributions, for 5.7 × 34.5 nm 4.5 ML NPLs and 7.5 × 33.2 nm 3.5 ML NPLs, respectively. At first glance, when performing the fits on samples with varying width, Δ_X shows no dependence on lateral confinement, with average values of 253 meV for 4.5 ML NPLs and 268 meV for 3.5 ML NPLs (Figure g,h, closed symbols), agreeing well with recent results obtained by two-photon absorption spectroscopy.

For the VQMC modeling of Δ_X (Figure g,h, open symbols, see also SI, Section S2), nonparabolicity of the bands should not be neglected; hence we use the thickness-dependent NPL tight-binding effective masses. We used a CdSe dielectric constant of ε = 10 and outer dielectric constant of ε _out = 2. We fitted the calculated data with an exponential decay ${\mathrm{\Delta }}_{\mathrm{X}}={\mathrm{\Delta }}_{\mathrm{X},\infty }·[1+A_{\mathrm{X}}\exp (-w/w_{0,\mathrm{X}})]$ , where A _X (= 0.52) is a global parameter for both thicknesses and w _0,X equals 3.4 and 4.2 nm for 3.5 and 4.5 ML NPLs, respectively (see also SI, Figure S2). Using the same A _X and w _0,X, and fitting the experimental data with only Δ_X,∞ as fit parameter, we obtain fair agreement between experimental and calculated values, with differences between experimental and calculated data of 12% and 21% for 4.5 and 3.5 ML NPLs, respectively. In principle, further agreement could be obtained by optimizing the dielectric constants of NPLs and surroundings and the effective masses of electrons or holes or by using a finite potential well in the calculations. More importantly, we can also observe that the fit essentially covers the scatter of Δ_X obtained experimentally, which is why we do not observe a dependence on the width. To reduce this scatter, the fit procedure of the absorption spectrum and corresponding extraction of Δ_X should likely be further refined.

Moving to Δ_BX, Figure a presents a schematic of the exciton and biexciton transitions in laterally confined NPLs. Biexciton formation in CdSe NPLs has been demonstrated using circularly polarized pump–probe spectroscopy. Considering transitions at K = 0, the exciton-to-biexciton (X → BX) absorption onset should manifest itself at energies E _BX,0 = 2E _X,0 – Δ_BX, with E _X,0 being the energy of the ground-state-to-exciton (0 → X) transition (Figure a). Depending on the state-of-motion of the exciton center-of mass, vertical transitions at K ≠ 0 can occur, and, since the biexciton is twice as heavy as the exciton, the different curvature of the exciton and biexciton states will lead to transitions at different K taking place at different energies. As such, the biexciton absorption spectrum is an asymmetric peak with a low-energy tail. In Figure b we report as a representative example the ΔA map of 3.5 ML CdSe NPLs with lateral dimensions of 6.4 × 41.1 nm, obtained by resonantly exciting the HH–E transition at a 10 μJ/cm² fluence (SI, Section S4). Alongside with the light- and heavy-hole exciton bleach signals, a photoinduced absorbance (PA) signal appears on the low-energy side, characterized by such a low-energy tail. The shape and spectral position of the maximum stay constant (Figure c). The spectral difference between the PA and the exciton bleach, equal to Δ_BX, is obtained by fitting the exciton bleach with a function that accounts for asymmetric broadening of the exciton line. , For the PA we used the same function convoluted with a Boltzmann factor, to include the thermal occupation of K ≠ 0 states for the biexciton , (SI, Section S4). A representative example of this fit is shown in Figure d for the ΔA spectrum recorded at 700 fs. However, the analysis can be simplified. Figure e shows the values of Δ_BX of the 3.5 ML NPLs obtained by fitting ΔA, together with Δ_BX calculated by directly taking the energy difference between the maximum of the PA and the minimum of the bleach (SI, Figure S3). Both values are in good agreement, highlighting that Δ_BX can be determined accurately by the simplified method.

3.

(a) Energy levels of the exciton (X) and biexciton (BX) states. The ground state-to-exciton (0 → X) and the exciton-to-biexciton (X → BX) transitions are indicated by the red and the blue solid arrow, respectively. Here E _X,0 and E _BX,0 are the exciton and biexciton energy at K = 0. E _g equals the free-carrier band gap, and Δ_X and Δ_BX are the exciton and biexciton binding energy, respectively. (b) Differential absorbance (ΔA) spectrum as a function of the pump–probe delay for 3.7 × 25.6 nm 3.5 ML NPLs, excited at 460 nm at a 10 μJ/cm² fluence. (c) Corresponding ΔA spectra at 700 fs, 2 and 3 ps. (d) Fit (gray) of the ΔA spectra (black) recorded at 700 ps and decomposition into biexciton PA (blue) and exciton (red) bleach signal. (e) Δ_BX of 3.5 ML CdSe NPLs with width ranging between 3.5 and 9 nm, obtained by fitting the ΔA spectra (closed circles) and by taking the energy difference between the bleach minimum and the PA maximum (open circles).

We thus evaluate Δ_BX for the set of CdSe NPLs investigated here using the energy difference between the PA and the exciton bleach. Figure a,b shows the width dependence of Δ_BX for the set of 3.5 and 4.5 ML CdSe NPLs investigated here (closed circles). The values of Δ_BX range between 45 and 55 meV and between 40 and 50 meV for 3.5 and 4.5 ML of CdSe, respectively. Notably, decreasing the thickness from 4.5 to 3.5 ML leads to an increase of about 10%, while decreasing the NPL width from 10 nm to 3 nm leads to a substantially larger increase of Δ_BX, by 30%. This highlights that reducing the lateral dimensions of the NPLs is a more robust way to enhance Δ_BX, compared with decreasing the thickness.

4.

Biexciton binding energies in CdSe NPLs. Experimental (full symbols) and simulated (open triangles, diamonds) Δ_BX of (a) 3.5 ML (blue) and (b) 4.5 ML (green) CdSe NPLs extracted from the ΔA spectra. The simulated data set is obtained with VQMC simulations, and ε = 10 for the exciton and ε = 9.5 for the biexciton (upward triangles), or ε = 10 for both (diamonds). (c) Ratio of the exciton-to-biexciton binding energies for 3.5 ML (blue) and 4.5 ML (green) CdSe NPLs. The Haynes rule is indicated by a dashed line.

In order to verify the experimental Δ_BX values, we again carried out VQMC simulations, under the following conditions: previous experiments and modeling on perovskite quantum dots , already showed that excitons and biexcitons polarize the lattice differently; hence, the effective dielectric constant that they experience can differ (see also SI, Section S5). We therefore calculated Δ_BX = 2·E _X – E _BX, from the exciton eigenenergy (E _X) using ε = 10, and the biexciton eigenenergy E _BX using ε = 9.5. We obtained good agreement between experimental and calculated data (Figure a,b) and confirmed that use of the different dielectric constants leads to an improved correspondence (compare to calculations using ε = 10 for both E _X and E _BX, open diamonds). We again fitted calculations with an exponential function, ${\mathrm{\Delta }}_{\mathrm{B}\mathrm{X}}={\mathrm{\Delta }}_{\mathrm{B}\mathrm{X},\infty }·[1+A_{\mathrm{B}\mathrm{X}}\exp (-w/w_{0,\mathrm{B}\mathrm{X}})]$ , where A _BX (= 0.81) is a global parameter for both thicknesses, and w _0,BX equals 4.6 and 6.1 nm for 3.5 and 4.5 ML NPLs, respectively. Using these parameters for the fit of the experimental data, Δ_BX,∞ , the biexciton binding energy of laterally unconfined NPLs extrapolates to 37 and 31 meV for 3.5 and 4.5 ML NPLs, respectively (see SI Figure S4 for extended width range).

Finally, now that we have obtained both Δ_X, from linear absorption spectra, and Δ_BX, from transient absorption spectra, we can determine the Haynes factor Δ_BX/Δ_X. A ratio of 0.228 was calculated by Singh et al. for an ideal (square) 2D biexciton geometry. In CdSe NPLs, however, due to their finite thickness and asymmetric electron and hole masses, biexcitons rather form a distorted tetrahedron. Figure c confirms that the experimental Haynes factor falls well below the theoretical limit for all samples. In agreement with previous results on epitaxial quantum wells, our data do not show a dependence on thickness. We do observe a decrease in the Haynes factor with increasing width, down to 0.15 for NPLs with large lateral sizes. This is due to the slightly different exponential decrease of Δ_X and Δ_BX with width (e.g., for 4.5 ML NPLs, w _0,X = 4.2 nm, w _0,BX = 6.1 nm), which can be understood from the larger size of the biexciton compared to the exciton, extending the width w for which lateral confinement still plays a role. At any rate, the experimental Haynes factor should provide an improved estimation of Δ_BX from Δ_X, if direct measurements are not available.

In conclusion, we carried out theoretical and experimental quantifications of the exciton and biexciton binding energy in CdSe nanoplatelets. Experimental Δ_X values only showed a small increase for thinner NPLs and no clear width dependence, while Δ_BX was observed to increase when reducing both the thickness and width. Variational quantum Monte Carlo simulations reproduce this behavior and confirmed that the small increase in Δ_X for laterally confined NPLs may fall within the experimental scatter observed. Since the increase of Δ_BX is significantly larger when reducing the width, our data highlight that lateral confinement, i.e., going to a nanowire geometry, is an efficient way to boost the biexciton binding energies in CdSe NPLs. In addition, results confirm that the Haynes factor in CdSe NPLs is lower than 0.228, obtained for an ideal square 2D geometry, with a value of 0.15 for larger NPLs. Our results provide insights into the fine-tuning of exciton and biexciton binding energies in colloidal NPLs and should be readily extendable to other classes of 2D nanomaterials, such as 2D perovskites, or transition-metal dichalcogenide nanosheets, or even hetero-NPLs, such as type-I CdSe/CdS core/crown NPLs, type-I CdSe/ZnS NPLs, or quasi-type II CdSe/CdS NPLs, where both charge delocalization and a modification of the dielectric constant by the semiconductor crown or shell can further influence the exciton binding energy. They can also provide valuable guidelines to engineer optoelectronic and photonic applications based on strongly bound excitons and biexcitons.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.nanolett.5c03251.

• Synthetic methods and structural analysis; details on the variational quantum Monte Carlo simulations; deconvolution of the absorption spectrum; details on the transient absorption spectroscopy and determination of the exciton and biexciton binding energy (PDF)

The authors declare no competing financial interest.