Localization-limited exciton oscillator strength in colloidal CdSe nanoplatelets revealed by the optically induced stark effect

2D materials are considered for applications that require strong light-matter interaction because of the apparently giant oscillator strength of the exciton transitions in the absorbance spectrum. Nevertheless, the effective oscillator strengths of these transitions have been scarcely reported, nor is there a consistent interpretation of the obtained values. Here, we analyse the transition dipole moment and the ensuing oscillator strength of the exciton transition in 2D CdSe nanoplatelets by means of the optically induced Stark effect (OSE). Intriguingly, we find that the exciton absorption line reacts to a high intensity optical field as a transition with an oscillator strength FStark that is 50 times smaller than expected based on the linear absorption coefficient. We propose that the pronounced exciton absorption line should be seen as the sum of multiple, low oscillator strength transitions, rather than a single high oscillator strength one, a feat we assign to strong exciton center-of-mass localization. Within the quantum mechanical description of excitons, this 50-fold difference between both oscillator strengths corresponds to the ratio between the coherence area of the exciton’s center of mass and the total area, which yields a coherence area of a mere 6.1 nm2. Since we find that the coherence area increases with reducing temperature, we conclude that thermal effects, related to lattice vibrations, contribute to exciton localization. In further support of this localization model, we show that FStark is independent of the nanoplatelet area, correctly predicts the radiative lifetime, and lines up for strongly confined quantum dot systems.


S1.2 Sizing of 4.5 ML CdSe Nanoplatelets
Transmission electron microscopy images of the 4.5 ML CdSe nanoplatelets used in this work are shown in Figure 1 of the main paper. The main analysis was done on this sample with average dimensions of 34×9.6nm 2 (= 326 nm 2 ). The histograms of this nanoplatelet's length and width are shown in Figure 1 below. Figure 2 shows TEM images of the other lateral areas used in this work, varying from 65 (20 x 3.4 nm 2 ) to 180 nm 2 (46.2 x 4.0 nm 2 ).

S1.4 Sample Treatment and Conditions
The samples were synthesized in air-free conditions and stored in a glovebox and in the dark.
For the measurements, fresh thin films were spincoated on glass substrates in ambient conditions and the experiments were also carried out in ambient conditions. As is well-known, experiments on nanoplatelets relying on generation of real charges (luminescence, gain, photodetection, ...) will be affected by effects such as oxidation or charging. However, Stark measurements explicitly do not produce any real charges in the material. By pumping below the band gap transition, we merely rely on an electric-field induced shift of the absorption Experimentally, the intrinsic absorption coefficient µ i of the platelets can be determined by measuring the absorbance in a cuvette and elemental analysis. Indeed µ i = ln (10)A f V ·L , where A is the absorbance, f V is the volume fraction of the platelets and L is the length of the cuvette.
However, as was shown by Achtstein et al. , it is also valid to calculate µ i experimentally, especially in a region where there is no difference between the intrinsic absorption of platelets and bulk material. Here, we choose to do this at 300 nm, similar to Achtstein et al.. 2 To do this calculation, Lorentz local field theory is used in combination with the Maxwell-Garnett effective medium approach. If the platelets are considered as oblate ellipsoids and assuming random orientation, the intrinsic absorption can be calculated as follows: where the local field factors are defined as follows i = (x, y, z): where s is the solvent permittivity and CdSe is the (complex) permittivity of Cdse nanoplatelets. Using the assumption of oblate ellipsoids, the depolarization factors L i can be calculated evaluating the following integral with x, y, z the half-lengths of the ellipsoid. To calculate the other polarization factors L x and L y , x, y and z need to be cyclically permuted.

S2.2 Quantum Well Absorption Spectrum
The following quantum-well absorption model is used to model the intrinsic absorption spectrum, see Figure 2): 3,4 with p X is the absorption line shape of a quantum well exciton with asymmetric broadening η: Here, E 0 and E x b are the absolute exciton energy and exciton binding energy, respectively.
Further, γ is the linewidth of the absorption peak, and the continuum edge has a step height of A C and a width of γ C . The total absorption α therefore becomes, including both light (LH) and heavy-hole (HH) contributions: with A HH and A LH the weight of the heavy-and light bands, respectively. A best fit is displayed in Table 1.
The binding energy of 193 meV can be translated into a Bohr radius as: Using m r = 0.085, we find a B = 1.5 nm for the 4.5 ML CdSe nanoplatelets used here. 4 HH LH

S2.3 Oscillator Strength of Platelets
Knowing the intrinsic absorption of the heavy hole exciton, the oscillator strength of the band gap can be calculated by integrating the heavy hole transition: 5 This integral can be directly used to calculate the oscillator strength: 5 where the local field factors are calculated in the same way as in section S2, but now around the band gap.
Takingg the case of the largest area sample (326 nm 2 ), we obtain local field factors (f x , f y , f z ) of (0.47,0.87,0.28), which leads to an average local field factor: |f The nanoplatelet volume is obtained from the TEM analysis as 1.37 × 34 × 9.5 nm 3 . The solvent refractive index is taken from hexane, n s = 1.5. Using the fitting values as shown in Table 1, we can reconstruct the function p X,HH which is then integrated on an energy scale. Finally, the oscillator strength F Abs of the heavy hole exciton is obtained: The table below lists the calculations for all the nanoplatelet areas used in this work.
Note that F abs,HH scales linearly with the nanoplatelet area.
S12  Figure S6 shows the fluorescence decay of the 4.5 ML CdSe nanoplatelets dispersed in nhexane after photo-excitation with a 400 nm pulsed laser diode at 2 MHz. We fit the decay of the integrated PL intensity with a triple exponential decay, similar to Leemans et al.
Similar to their work, we assign the fast component τ 1 to radiative decay, which amounts to 65 percent of the total decay, a number line with the quantum yield of the sample indicating a heterogeneous population of emitters. Given the more complex decay pattern in our core-only nanoplatelets, we decided to assign an average lifetime as:  Figure S6: Fluorescence decay after photo-excitation at 400 nm with a 2 MHz pulse train.

S4 Occurrence of Two-Photon Absorption
Upon strong sub-band gap excitation, also real carriers can be generated through 2-photon absorption (2PA). 7 After a 2PA event, a carrier pair is created with a large excess energy.
For example, 2PA from a 580 nm pump would create a carrier pair equivalent to a 4.3 eV excitation. We observe such effects at higher pump power as evidenced in Figure S7 where a clear long-lived bleach associated with the creation of real electron-hole pairs is obtained at high pump fluence.

S5 Polarization Analysis
If we assume a random orientation of the nano-platelets, we can write the absorbance of the thin film A in general as: where f i is here taken as the absolute value of the local field factor, as defined in section S2. We note that it needs to be evaluated at the position where the OSE is evaluated (514 nm). The absorbance A 0 is that of unscreened nanoplatelets. Figure S8: Orientation effect on polarization analysis, overview of notation.
We measure an energy shift ∆E which is an average over all orientations of the nanoplatelets.
Given this random orientation of the platelets, we can write: If we replace A 0 for A using the expression above, we obtain: The equations above define the factors f xx and f yy as used in the manuscript: The average electric field F is linked to the average intensity I 0 through: where n is the refractive index. 0 c are the vacuum permittivity and speed of light respectively. 8 The intensity is calculated as Ep A×tp where t p = 120 fs is the pulse width, A is the beam area and E p is the energy per pulse.

S7.1 Electron-Hole Pair States as a Basis
To discuss exciton localization and the exciton oscillator strength, we take the approach that states in a crystalline solid can be described as the product of a Bloch wavefunction and an envelope wavefunction, the latter being a solution to the effective mass Schrödinger equation. In that case, the unperturbed single-particle states in the conduction and valence band correspond to free particle waves characterized by wavevectors k e and k h . In coordinate representation, where r e and r h are the coordinates of the electron and the hole, respectively, the corresponding states can be written as: Here, n referst to the dimensionality of the material at hand.
To describe a general electron-hole pair state -such as a bound exciton or a localized exciton -the free particle product states | k e , k h = | k e | k h can be taken as a basis. In that case, each electron-hole pair state | X can be expanded as: Here, the expansion coefficient Φ X (k e , k h ) is introduced as a mere representation of the inner product k e , k h | X . A similar starting point was put forward by Dresselhaus and by Elliott. 9,10 To make Eq S23 more tangible, one can re-express the inner product k e , k h | X by means of the completeness relation in the electron and hole coordinates r e and r h . We thus S19 obtain: Here, the inner product can be made explicit using Eqs S21 and S22, whereas r e , r h | X is the wavefunction Ψ p (r e , r h ) of the electron-hole pair state | X . Accordingly, we can rewrite the expansion coefficient Φ X (k e , k h ) as: Hence, we retrieve the familiar result that the expansion coefficients of a quantum state | X in coordinate and wave vector representation are coupled by a Fourier transform; a result that is well known from free particle quantum mechanics. Using the properties of Fourier transform pairs, we also have:

S7.2 The Exciton Wavefunction
The exciton state in a perfect crystal is characterized by an envelope wavefunction Ψ(r e , r h ) that is an eigenfunction of a hydrogen-like Hamiltonian, which describes the electron-hole pair bound by a screened Coulomb interaction: Here, all symbols have their usual meaning.
This Schrödinger equation is conveniently solved by a transformation from the electron and hole coordinates r e and r h to coordinates R and r that describe the center-of-mass of S20 the electron-hole pair and the electron-hole interdistance: Importantly, the concomitant wavevectors K and k read: 1 It is well known that with this transformation, the solutions Ψ(r e , r h ) to the Schrödinger equation are obtained as: 10 Here, the center-of-mass motion is described as a plane wave, whereas the internal motion corresponds to a bound or unbound hydrogen-like wavefunction χ ν (r), as characterized by the set of quantum numbers ν. 2

S7.3 The Matrix Element for Interband Optical Transitions
Following Elliot, 10 we use the expansion of a general electron-hole state | p as expressed by Eq S23 to write the transition matrix element X | H | 0 of the light-matter interaction operator H = E · µ as: 3 This expression recognizes the fact that the actual transition from the ground state | 0 to the electron-hole pair state | X involves the promotion of an electron from the valence band state | − k h to the conduction band state | k e . Moreover, the subscript B indicates that the Bloch states and not the envelopes must be used for the calculation of the matrix element In coordinate representation, the matrix element k e,B | H | − k h,B becomes an integral over the r e coordinate: To evaluate this matrix element, we introduce the following ideas: • For k e = k h = 0, the integrandum will have the full periodicity of the lattice. Hence, the matrix element 0 | H | 0 can be calculated as an integral over a single unit cell, normalized by the volume of a unit cell. Hence, it is a number independent of the volume of the semiconductor.
• If either of both wavevectors differs from 0, the integrandum is the product of a part with the full periodicity of the lattice, and a wavelike part with wavevector k e + k h . 4 When the crystal is sufficiently large, the matrix element will therefore vanish unless k e +k h = 0. This is the well-know rule that only interband transitions that are vertical in reciprocal space are allowed.
In what follows, we will assume that the relevant wavevectors involved in the interband transitions are all small, such that the matrix element k e,B | H | − k e,B , which is weakly dependent on k e can be replaced by c 0 | H | v 0 . Here, | c 0 and | v 0 represent the conduction and valence-band states at k = 0. Using this substitution, Eq S33 can be rewritten 4 In principle, the matrix element adds the wavevector of the electromagnetic field as an additional component here, but this we neglect since electron wavevectors will be much larger than photon wavevectors.

S7.4 Transitions to Delocalized, Bound Excitons
With the factorization of Ψ(r e , r h ) into a center-of-mass wavefunction and an internal wavefunction, the Fourier transform linking wavefunction and expansion coefficients Φ(k e , k h ) can be rewritten as: Here, ψ(R) represents a general center-of-mass wavefunction. For the case that K = 0, and assuming free-particle motion for the center-of-mass, this yields: Reversing this relation by using the properties of the Fourier integral, we also have: 5 Evaluating χ ν at r = 0, we thus obtain: Referring to Eq S35, we can express the summation over k e as an integral by considering that a wavevector range dk e will contain (L/2π) n × dk e different wavevectors. Hence: As a result, the transition matrix element to form a delocalized exciton in the bright | Φ K=0 state is obtained as: Accordingly, we obtain the oscillator strength F K=0 for the formation of a 2D delocalized exciton as: 6 We thus conclude that the oscillator strength for the formation of delocalized excitons in the | Φ K=0 scales proportionally with the area S of the quantum well.

S7.5 Transitions to Localized, Bound Excitons
Assuming that the state | X of a localized, bound exciton can still be described as the direct product of the center-of-mass part and the internal part, Eq S43 yields: In the case of a 2D system, the first integral has units of length, and its square is defined as the coherence area S coh of the exciton center-of-mass: We thus conclude that the ratio between the oscillator strength to form a localized and a delocalized amounts to the ratio S coh /S between the coherence area and the total area of the quantum well:

S7.6 The Coherence Area of a Gaussian Wavepacket
In the case of a two-dimensional Gaussian Wavepacket, the center-of-mass motion wavefunction ψ(R) can be written as: This yields the coherence area as:

S8 Low Temperature Measurements
For the low temperature experiments, samples were again spin-coated on transparent glass slides and mounted in a liquid nitrogen cooled cryostat. Temperature was first ramped down to 77K and gradually increased to room temperature. A pump at 700 nm, 600 meV detuning, was used to initiate the OSE.  to the synthesis mixture at a rate of 12 mL/h to keep the reaction rate at a high level, which enables to grow large CdSe. The reaction was stopped by thermal quenching using a water bath after approximately 2 h. The dark red reaction mixture was purified by the addition of toluene, isopropanol and methanol, in a 1:1:1 ratio relative to the volume of the reaction mixture. The resulting turbid solution was centrifuged to obtain a precipitate of QDs that was re-dispersed in toluene. Next, purification was repeated three times using toluene and acetone as solvent and non-solvent, respectively, to remove all residual reaction products.

S10.2 Intrinsic Absorption Coefficient of CdSe Quantum Dots
The intrinsic absorption coefficient of spherical zinc blence CdSe QDs can be calculated in a similar manner as for nanoplatelets. The depolarization factors amount to 1/3 in case of a sphere, hence simplifying the formula for µ i to: with f the local field factor for a spherical inclusion: We calculate µ i at 320 nm, where |f (320)| 2 = 0.228, using CdSe = 8 + 6.6i. This amounts to µ i (320) = 1.65 × 10 5 cm −1 . Figure S2b shows the absorption spectrum normalized to represent µ i .

S10.3 Oscillator Strength
Using the same approach as for the CdSe platelets lined out in S2.3, we can calculate the oscillator strength of the band edge transition in the 6.25 nm CdSe QDs using the integrated intrinsic absorption spectrum: The local field factor is calculated using the expression for spherical inclusions, i.e. L i = 1/3 in the framework of the depolarization factors lined out earlier. We obtain a value of: |f LF | 2 = 0.287 using bulk dielectric constants of c-CdSe at 1.93 eV, 1.93eV = 8 + 1.6i, and a solvent dielectric constant of 2.25 (toluene, n s = 1.5). 11 µ i,gap is calculated from the intrinsic absorption spectrum, see Figure S2, by integrating from +∞ to 1.93 eV and doubling the value. We obtain f abs = 11.9, in line with the values calculated by Moreels et al. 12 Figure S13a shows the 2D false colour map of ∆A after a 700 nm pulse (∆ =160 meV). Figure   S13b shows a kinetic slice taken at 650 nm. We fit a rising background to accomodate for the creation of real carrier through 2PA (2-photon absorption). Note that this is not required at lower densities where 2PA is negligible.