Nature and Control of Shakeup Processes in Colloidal Nanoplatelets

Recent experiments suggest that the photoluminescence line width of CdSe and CdSe/CdS nanoplatelets (NPLs) may be broadened by the presence of shakeup (SU) lines from negatively charged trions. We carry out a theoretical analysis, based on effective mass and configuration interaction (CI) simulations, to identify the physical conditions that enable such processes. We confirm that trions in colloidal NPLs are susceptible of presenting SU lines up to one order of magnitude stronger than in epitaxial quantum wells, stimulated by dielectric confinement. For these processes to take place trions must be weakly bound to off-centered impurities, which relax symmetry selection rules. Charges on the lateral sidewalls are particularly efficient to this end. We propose that the broad line width reported for core/shell CdSe/CdS NPLs may relate not only to SU processes but also to a metastable spin triplet trion state. Understanding the origin of SU processes opens paths to rational design of NPLs with narrower line width.


Results
We analyze the emission spectra of trions in core-only and core/shell NPLs. Negative trions are studied unless otherwise noted, as it is the most frequently reported species in these structures, but the conclusions do not depend on the sign of the charged exciton (see Fig. S2 in the supporting information, SI). Once the general behavior of SU processes in these systems is understood, we discuss how our conclusions fit the interpretation of different experimental observations and the practical implications of our findings.

Core-only NPLs
We start by studying core-only CdSe NPLs. The NPLs are chosen to have 4.5 monolayer (ML) thickness and a lateral size of 20 × 20 nm 2 , for similarity with the core dimensions of Ref. 20 They have a pronounced dielectric mismatch with the organic environment, which we model with ǫ in = 6 and ǫ out = 2 as dielectric constants inside and outside the NPL, unless otherwise stated. 28,29 The presence of few-meV spectral jumps in photoluminescence experiments 20 suggests that the trion is subject to the influence of carriers temporarily trapped on the surface. 19,30 To model this phenomenon, a fractional point charge is placed on the surface, with charge Q = e Q X (|Q X | ≤ 1 and e the full electron charge). The fractional value of Q X accounts for the screening of trapped charged (e.g. hole) by the trap defect itself (e.g. surface dangling bond). 31 Two scenarios are considered: a charge centered on the top facet (Q top ) and an off-centered charge, located along the edge of a lateral facet (Q edge ). The latter setup is suggested by studies showing that edge and vertex atoms in CdSe structures have weaker binding to oleate ligands. 32 The two systems are represented in Figure 1a Fig. 1c), but their strength is two orders of magnitude smaller than that of the fundamental transition (main line). This is similar to the case of epitaxial quantum wells. [22][23][24][25] (iii) Stronger SU replica are however obtained for charges located on the lateral sidewall, provided the charge is attractive (acceptor impurity) and binding to the trion is moderately weak, see Fig. 1d. For Q edge = 0.4 (marked with a star in the figure), the SU peak reaches ∼ 25% of the main peak height. This ratio is about 20 times higher than in epitaxial quantum wells, and it holds despite the Giant Oscillator Strength enhancing the band edge recombination, [5][6][7]29 which suggests that SU satellites also benefit from this phenomenon. For Q edge > 0.4, however, the SU peak intensity is lowered again and the energy splitting (redshift) with respect to the main line increases. Second (c,d) Corresponding X − emission spectrum for charge strength Q = Q X e. The arrows point at the SU satellites (dotted lines are guides to the eyes). The highest SU peak is observed for off-centered acceptor charges weakly bound to the trion (Q edge = 0.4, marked with a star in (d)). The spectra are normalized to the intensity of the fundamental transition at Q X = 0, and offset vertically for clarity. The insets for Q edge = 0.7 in (d) show amplified SU peaks.
To gain understanding on the origin of strong SU peaks when trions bind to lateral surface acceptors, beyond the full numerical calculation of Fig. 1, in Fig. 2a and 2b we compare sketches of the SU processes, in the absence and presence of an attractive edge charge. Within effective mass theory, the conduction band and valence band energy levels of (non-interacting) electrons and holes can be described as particle-in-the-box states, with quantum numbers (n x , n y , n z ). It is useful however to label the states by their symmetry (irreducible representation). When Q edge = 0, because the NPL has squared shape, the point group is D 4h . When Q edge = 0, the electrostatic potential yields a symmetry descent to C s .
As a consequence, degeneracies are lifted and additional states with the same symmetry as the ground orbital (A ′ ) are obtained. This is important because after electron-hole recombination, the excess electron can only be excited to an orbital with the same symmetry as the initial one (vertical arrows in Fig. 2a and 2b). Therefore, lowering the system symmetry opens new channels for SU processes. Furthermore, these can involve low-energy orbitals, which have fewer nodes and will then have larger overlap with the trion ground state, as we shall see below. Both the number and the intensity of the SU processes are in principle enhanced. By contrast, a centered charge on the top surface barely affects the system symmetry, which remains high (C 4v ), and SU processes are only slightly stronger than in the Q edge = 0 case. The qualitative reasoning above can be substantiated with a CI formalism on the basis of independent particle (non-interacting) electron and hole states, which has the additional advantage of giving intuitive insight on how Coulomb interactions affect the likelihood of SU processes. We consider that the transition rate from the trion ground state |GS X − to an electron spin-orbital |f e , is proportional to: 33 (1) P is the dipolar transition operator,P = ie,i h i e |i h e ie h i h , where e ie and h i h are annihilation operators for independent electron and hole spin-orbitals |i e and |i h , respectively. We   Figure 2: (a,b) Sketch of SU processes in NPLs with (a) and without (b) an edge charge. Labels on the left are (n x , n y , n z ) quantum numbers for the (independent particle) energy levels. Labels on the right are the corresponding irreducible representation. The surface charge lowers the point group symmetry, from D 4h to C s , lifting degeneracies and enabling new channels for SU transitions (vertical arrows). (c,d) Two main configurations |m X − in the CI expansion of |GS X − , with and without edge charge. Thin (thick) arrowsheads denote electron (hole) spin. Only when Q edge = 0 a SU process is expected. (e) Energy splitting between |1 X − and |2 X − at an independent particle level. (f) average value of electronelectron repulsion and (g) electron-hole attraction in configurations |1 X − and |2 X − . describe the trion ground state with a CI expansion, where |m X − is a trion configuration: |m X − = e † re e † se |0 e h † t h |0 h , with e † re and h † t h creator operators, |0 e and |0 h the vacuum occupation vectors of electron and hole, and c m the coefficient in the expansion. InsertingP and |GS X − into Equation (1), one obtains: ( In SU processes, |f e is an excited spin-orbital. It then follows from Equation (3) that such a transition will only take place if |GS X − contains at least one configuration |m X − in the CI expansion where one electron is in the excited spin-orbital and the other electron has finite overlap with the hole ground state (|s e = |f e and r e |t h = 0 or |r e = |f e and s e |t h = 0).
The larger the weight of this configuration, |c m | 2 , the more likely the SU process. It is worth noting that in the strong confinement limit, the trion ground state is well described by a single configuration where all carriers are in the lowest-energy spin-orbitals (configuration |1 X − in NPLs constitute an ideal system at this regard, because they combine weak confinement in the lateral direction with strong Coulomb interactions. 34,35 Hereafter, we refer to this condition (c m = 0 for m > 1) as Coulomb admixture.
The role of Coulomb correlation and symmetry breaking in activating SU processes can be illustrated, in the simplest approximation, by considering the two lowest-energy configu-rations of the trion ground state, In Fig. 2c and 2d we depict such configurations in the absence and presence of an edge charge, respectively. These can be expected to be the two most important configurations in the full CI expansion. Notice that the two configurations must have the same symmetry, for Coulomb interaction to couple them. Because the lowest-energy configuration, |1 X − , is always totally symmetric, so must be |2 X − . Thus, when Q edge = 0 (D 4h group), the electronic configuration The recombination of the E u electrons with the hole, which stays in a A 1g orbital, is then symmetry forbidden ( r e |t h = s e |t h = 0 in Eq. (3)). By contrast, when Q edge = 0 (C s group), |2 X − is formed by a monoexcitation where one electron is placed in the (n x , n y , n z ) = (2, 1, 1) orbital, which also has A ′ symmetry, resulting in an electronic configuration [ Fig.2d). The hole can then recombine with the ground orbital electron, as both have A ′ symmetry ( r e |t h = 0 or s e |t h = 0 in Eq. (3)) and leave the excited electron as the final state. This constitutes a SU process. Because both SU and fundamental transition rely on the recombination of the same electron-hole pair (same overlap integral, e.g. r e |t h ), the ratio between SU and fundamental radiative rates can be approximated as: i.e. it is set exclusively by the degree of Coulomb admixture.
One can guess the requirements that maximize |c 2 | 2 by looking which conditions favor energetically |2 X − over |1 X − . These include: (i) small energy splitting between the two configurations, at an independent particle level, ∆ sp in Fig. 2d, (ii) weaker electron-electron repulsion (V ee ) and (iii) stronger electron-hole attraction (V eh ) in |2 X − as compared to |1 X − . When the off-centered charge is switched on, ∆ sp rapidly decreases (see Fig. 2e) because the symmetry descent turns one of the E u (p-like) electron orbitals into a A ′ (s-like) one. However, the surface charge brings about electrostatic confinement and hence ∆ sp increases again soon after. As for inter-electron repulsion, 1 X − |V ee |1 X − increases more rapidly than 2 X − |V ee |2 X − (see Fig. 2f) because the former involves placing the two electrons in identical orbitals, while the latter does not. Last, 1 X − |V eh |1 X − is rapidly quenched (see Fig. 2g) because it involves the ground orbitals of electron and hole -(1, 1, 1) e and (1, 1, 1) h -, which dissociate rapidly under an external charge. 2 X − |V eh |2 X − stays strong up to Q edge ∼ 0.3 because it involves the (2, 1, 1) e orbital, which is spatially more extended and then keeps significant overlap with the (1, 1, 1) h hole. Figs. 2e-f further evidence that Q edge > 0.3 − 0.4 is inconvenient for SU processes, because the electrostatic potential increases lateral quantum confinement (∆ sp increases) and because electrons and At Q edge ≈ 0, the two orbitals are quasi-orthogonal. As a result, Coulomb interaction cannot couple configurations |1 X − and |2 X − . and c 2 ≈ 0. This is why the two-electron charge density closely resembles the (1, 1, 1) e orbital. SU processes are not expected in this case.
At Q edge ≈ 0.4, symmetry lowering and energetic considerations enable efficient Coulomb coupling. The oval shape of the two-electron charge density reflects a significant contribution from (2, 1, 1) e to |GS X − (i.e. |c 2 | > 0). At the same time, the electron (1, 1, 1) e orbital and the hole ground state have sizable overlap. This is an optimal situation for the appearance for the transition P GS→(2,1,1)e to show up as a SU process, according to Equation (3). Further increasing Q edge separates the (2, 1, 1) e electron orbital from the hole. Coulomb attraction is then weaker, making c 2 and consequently P GS→(2,1,1)e small again. Figure 4: Normalized X − emission as a function of the environment dielectric constant. With increasing dielectric contrast, the SU peak increases and becomes more redshifted. For every value of ǫ out , the value of Q edge that maximizes SU transitions is shown. In all cases, ǫ in = 6.
We have argued above that strong Coulomb admixture of configurations facilitates the appearance of SU processes. A distinct feature of colloidal NPLs when compared to epitaxial quantum wells is the presence of a prounounced dielectric contrast with the organic ligands surrounding the NPL, which enhances Coulomb interactions by effectively reducing the system dielectric screening. 29,34,36 To study the influence of this phenomenon over SU transitions, in Figure 4 we compare the trion emission spectrum for different values of the environment dielectric constant ǫ out , while fixing that of the NPL to the high-frequency CdSe value, ǫ in = 6. For the sake of comparison, the emission spectrum is normalized so that the band edge peak has the same intensity in all cases. Also, we have selected the value of Q edge that maximizes the relative size of the SU peak in each case. Because ǫ out screens the surface charge electrostatic field, larger Q edge values are needed when ǫ out increases. The figure evidences that lowering ǫ out increases the SU peak height and energetic redshift. For typical ligands of CdSe NPLs (e.g. oleic acid), ǫ out ∼ 2. 29, 37 We then conclude that dielectric confinement makes SU processes in colloidal NPLs more conspicuous.

Core/shell NPLs
We next consider heterostructured core/shell NPLs. The first case under study are CdSe/CdS NPLs. [12][13][14]38 The NPLs have the same CdSe core as in the previous section and 6 ML thick CdS shells on top and bottom (see inset in Figure 5a). In general, the behavior of SU replicas is found to be analogous to that of core-only NPLs. An off-centered acceptor impurity is needed to yield sizable SU replicas, with an optimal value of Q edge maximizing the relative size of the SU peak. Figure 5a shows the emission spectrum of X − for the optimal Q edge value, in CdSe/CdS NPLs (green line) against CdSe core-only NPLs (black, dashed line). One can see that the SU replica of the CdSe/CdS structure is again significant (11% of the main transition), but less pronounced than in the core-only structure (26% -15 15 30 Qedge  Understanding the conditions which promote SU processes allows us to devise structures where their impact would be maximal. In Fig. 6 we consider a core/shell NPL with the same dimensions as before, but CdSe/CdTe composition. The NPL is chosen to be charged with a positive trion (X + ). Because of the type-II band alignment, the electron stays in the CdSe core and the holes in the CdTe region, as observed in related core/crown structures. 39,40 In the absence of external charges, the two first hole orbitals are (1, 1, 1) h and (1, 1, 2) h , i.e. the symmetric (A 1g ) and antisymmetric (A 1u ) solutions of the double well potential, respectively, which are almost degenerate because tunneling across the core is negligible (i.e. ∆ sp → 0).
Switching on a negative surface charge, Q edge < 0, lifts the inversion symmetry so that both orbitals acquire A ′ symmetry and can be Coulomb coupled. The admixture between configurations |1 X + and |2 X + , depicted in Fig. 6b, is then very strong. In the presence of the charge, the two hole orbitals tend to localize on opposite shell sides to remain orthogonal, as shown in Fig. 6c. This implies that configuration |1 X + , which has two holes in the same orbital, has much stronger repulsion than configuration |2 X + , which distributes the two electrons on opposite sides of the core. This makes 1 X + |V hh |1 X + ≫ 2 X + |V hh |2 X + .
Altogether, the small ∆ sp value and the large difference in hole-hole repulsion explain the strong admixture between configurations |1 X + and |2 X + . As shown in Fig. 6a, this gives rise to SU peaks whose magnitude is almost as large as that of the fundamental transition (72% for Q edge = −0.5).

Discussion
Our simulations show that SU processes can be expected for trions in core-only and core/shell NPLs, if off-centered impurities are present. We discuss here the potential relationship of this finding with experiments and practical implications.

Relationship with experiments
In core-only CdSe NPLs, the low temperature photoluminescence is thought to arise from subpopulations of excitons and negative trions.  Fig. 1b). This possibility is suggested by studies showing that Z-type ligand desorption -and hence surface traps-in CdSe NPLs is more frequent on these facets, 44 and by the fact that CdSe/CdS core/crown NPLs generally improve the photoluminescence quantum yield as compared to core-only structures, despite having larger surfaces on top and bottom. 45 Because off-centered charges are needed to originate SU peaks, lateral charges are candidates to trigger such processes.
In core/shell CdSe/CdS NPLs, SU processes have been also proposed as the origin of multi-peaked fluorescence emission -and hence broadened line width-. 20 Our simulations in Fig. 5a confirm one can indeed expect a sizable SU peak in such structures. We note that earlier experimental studies had so far interpreted the line width broadening as a result of either SU processes 20 or of surface defects. 12 By showing that the second effect is a prerequesite for the first one, our study helps to reconcile both interpretations. Nonetheless, two remarkable disagreements are observed between our simulations and Ref. 20 measurements.
First, the experiments show from 2 to 4 emission peaks, which are interpreted as the X − fundamental transition plus up to three redshifted, SU peaks. In our calculations, however, we fail to see more than one significant SU replica. Second, the highest-energy peak in the experiment is never the brightest one. This is inconsistent with our results and with earlier studies on epitaxial quantum wells and dots, where the higher-energy peak corresponds to the fundamental transition, which is the most likely recombination channel. [22][23][24][25][26] Tentatively, one may suspect that a large number of SU peaks in core/shell CdSe/CdS NPLs could be connected with the thick CdS shell (12 ML in Ref. 20 ), which makes surface defects more likely than in core-only structures. A significant presence of defects in these structures has been hinted by studies showing that the long radiative lifetime is not due to electron delocalization but to the influence of impurities. 13 However, Coulomb interactions are weaker than in core-only structures (Fig. 5c,d), where only one SU peak has been measured. 21 It is then not surprising that, despite investigating different charge locations (Figs. S3, S6 and S7 in SI), conduction band-offset values (Fig. S4) and shell thicknesses (Fig.S5), we see at most one significant SU satellite.
Regarding the relative intensity of the peaks, as mentioned in the previous section, the highest-energy one (fundamental transition) is proportional to the weight of configuration |1 X − in the CI expansion, |c 1 | 2 , while subsequent (SU) peaks would be proportional to |c 2 | 2 , |c 3 | 2 , . . . Configuration |1 X − (all carriers in the ground orbital, Fig. 2c) is nodeless and hence naturally expected to be the dominant one, so the highest-energy peak is also the brightest one. We have not observed SU peaks exceeding the fundamental transition height despite considering different charge locations and shell thicknesses (see SI). Even in CdSe/CdTe NPLs, which constitute a limit case, SU peaks never exceed the height of the main transition, see Fig. 6a. band edge recombination To illustrate this point, in Figure 7a we show the calculated emission of X − assuming equipopulation of S e = 0 and S e = 1 trion states. One can see that the number of sizable peaks in the spectrum ranges from two to four, depending on the strength of surface charge, Q edge . The origin of these peaks is summarized in the sketches of Fig. 7b and 7c. The singlet ( Fig. 7b) can give rise to a fully radiative transition (s-R1) and a SU transition (s-SU), as described in the previous sections. In turn, the triplet (Fig. 7c)  For example, because all peaks in Fig. 7a arise from the same NPL, they will experience simultaneous spectral shifts when surface impurities migrate. 20 Also, the hot trion emission is expected to vanish when the impurities are removed, as t-R1 becomes deactivated and t-R1 almost merges with the singlet emission, s-R1, see Fig. 7a for Q edge = 0. This fits the transition from asymmetric to symmetric band shape as temperature increases. 12 The fact that triplet emission is observed in CdSe/CdS NPLs, but not in CdSe ones, may be explained from the strong spin-spin interaction of resident carriers and surface dangling bonds in the latter case, 48 which should speed up spin relaxation through flip/flop processes.
This mechanism is expected to be inhibited in core/shell structures, because X − carriers stay far from the surface, as shown in Fig. 5d. On the other hand, the triplet trion is expected to have fine structure through electron-hole exchange interaction, 49 which may not fit the mono-exponential photoluminescence decay reported in Ref. 20 Further experiments are needed, e.g. on polarisation of the different peaks under external fields, 41,50 to confirm the different spin of the emissive states in CdSe/CdS NPLs.
The observation of metastable triplet trion photoluminescence has been previously reported in epitaxial quantum wells 24,51 and dots, 50 and more recently in transition metal chalcogenide monolayers. 52 To our knowledge, however, its presence in colloidal nanostructures has not been confirmed.

Control of SU processes
Inasmuch as SU processes can be responsible for the line width broadening NPLs, their supression is desirable to improve color purity in optical applications. It has been suggested that this job could be achieved by increasing quantum confinement, reducing either lateral dimensions or shell thickness -the latter would favor electrostatic confinement. 20 Both strategies have the drawback of introducing size dispersion in ensemble luminescence. From our theoretical analysis, we confirm that reducing Coulomb admixture would minimize SU processes, but this can be achieved by weakening Coulomb interactions instead of increasing quantum confinement. For example, reducing dielectric confinement or using thinner cores to enhance the quasi-type-II character should contribute to this goal. Obviously, this approach would have the drawback of reducing the band edge recombination rate as well.
Alternatively, since our study shows that impurities are ultimately responsible for SU processes, experimental routes to suppress SU processes could be directed to control of traps. Appropriate choice of surface ligands, 44 electrochemical potentials 53 and interface alloying 16,17 could contribute to this end.
Because we find surface charges on lateral sidewalls particularly suited to induce SU processes, the growth of core/crown heterostructures is expected to reduce their influence by keeping the outer rim away from the photogenerated carriers. This suggestion seems to agree with experimental observations by Kelestemur and co-workers, indicating that core/crown/shell CdSe/CdS NPLs have more symmetric emission behavior than core/shell ones at cryogenic temperatures, 54 This can be understood as a consequence of the suppression of SU processes in the low-energy tail of the emission band. It is also consistent with This is at least one order magnitude larger than in epitaxial quantum wells. The SU peak is redshifted from the band edge peak by up to few tens of meV, thus providing a source of line width broadening.
These results are in excellent agreement with recent experimental findings in CdSe NPLs 21 in terms of number of emission peaks, energy splitting and relative intensity, but only partially so with those of core/shell CdSe/CdS NPLs. 20 Experiments in the latter structure are however in line with an alternative interpretation involving simultaneous participation from trion singlet and metastable triplet states.
Strategies to narrow the line width of NPLs through suppression of SU processes should aim at controlling electrostatic impurities or Coulomb admixture.

Additional calculations
We present here additional calculations for further understanding of SU processes.

Convergence of CI calculations
Configuration Interaction (CI) calculations on the basis of independent particle (or Hartree-Fock) orbitals provide an excellent description of repulsions in few-and many-fermion systems. 1,2 However, large basis sets are needed to describe strong attractions, 3,4 which are certainly present in colloidal NPLs 5 and are involved in a correct description of SU processes. Figure S1: X − emission spectrum for Q edge = 0.4 (see main text). Zero energy is set for the fundamental transition with ne = nh = 22. ne and nh are the number of single-electron and single-hole spin-orbitals, respectively, used to build the CI basis sets.
In Fig. S1 we compare the X − emission spectrum calculated for CdSe NPLs -same dimensions as in main text-using different basis sizes. The basis is formed by all possible combinations of the first ne (nh) independent particle spin-orbital states of electrons (holes).
With increasing basis dimensions, the band edge transition peak redshiftes and gains intensity, which reveals an improved description of electron-hole correlation. The intensity of the SU peak height and its redshift with respect to the band edge transition are however less sensitive to the basis dimensions. It follows from the figure that quantitative assessment on S2 the ratio of fundamental vs SU peak heights requires large basis sets. In the main text we use ne = nh = 22. By comparing with smaller values of ne/nh in the figure, it is clear that for this value -which involves very time-consuming computations-the ratio is reaching saturation. This validates the order of ratios provided in the main text. For the calculations in this Supporting Information, however, we may resort to ne = nh = 12, which overestimates the relative height of SU peaks, but suffices to provide qualitative assessment.
Positive trion behaviour -20 -10 0 10 20 Emission (a.u.) Figure S2: X + normalized spectrum emission for different charge intensities. The arrows are pointing to SU satellites (dotted lines are guides to the eyes). The highest SU peak (Q edge = −0.3) is marked with a star. The origin of energies is set at the band edge recombination peak. The insets correspond to Q edge = 0.5 amplified SU peaks.
In the main text, we have mostly considered the case of negative trions. We show here that the same behavior holds for positive ones. To illustrate this point, we choose the case of the core-only NPL with an edge charge, equivalent to Fig.1d of the main text. Figure S2 shows that the presence of SU peaks in the emission spectrum is again strongly dependent on the value of the surface charge. For Q edge = 0, no SU peak is observed. For repulsive (Q edge > 0) charges, SU are formed but very small in magnitude. The highest SU peaks are formed for weakly bound donor charges (Q edge < 0), which attract the holes of X + , marked with a star in the figure. As in the X − case, if the attractive charge further increases it S3 starts dissociating the trion. Consequently, SU peaks are quenched again. Notice however energy splittings for X + (Fig. S2) are smaller than for X − (Fig. 1d in the main text). This is expected from the heavier masses of holes. In the main text we present the representative cases of a surface charge centered on top of the NPL (Q top ), and centered and that of a charge on the edge of lateral sidewall (Q edge ).

Effect of charge impurity location
In Figure S3 we compare with different locations. One can see that the effect of a charge located in the corner, red line in Fig. S3a, provides similar SU peaks to that of the edge charge, blue line in the figure, both in energy and intensity. We recall that these traps seem to be particularly likely according to recent studies on ligand desorption. 6,7 Off-centered charges on top and bottom surfaces are studied in Fig. S3b. They give rise to SU peaks of similar height to that of Q edge , although they reach the optimal charge value sooner than Q edge (Q top−edge ∼ Q top−corner ≈ 0.2 versus Q edge = 0.4), because they lie closer to the center of the NPL, where photogenerated carriers tend to localize.

Effect of conduction band offset in CdSe/CdS NPLs
The value of the CdSe/CdS conduction band offset (CBO) has been a subject of debate in nanocrystal heterostructures. [8][9][10] We used, along our main text, an upper-bound unstrained value of 0.48 eV, 8 which is partly reduced by compressive strain in the core. 10 Here we explore the scenario where we use a lower-bound 9 value as well, to see the possible effect of enhancing electron delocalization over the CdS shell. Figure S4 compares the two cases.
Lowering the CBO gives rise to slightly weaker electron-electron repulsion (V ee ) and electronhole attraction (V eh ), however the differences are very small. One can then expect similar role of SU processes as in the main text.

b)
Qedge In this section we compare qualitatively the response in the two cases using a moderate basis set (ne = nh = 12), which permits addressing the experimental dimensions without the computational burden of the large basis set (for 12 ML thickness, the extended CI computation is beyond our current resources). 0.5 If we focus on the charge location in both systems, Fig. S5a, one may expect similar behaviour. The main difference, as can be seen in Fig. S5b (left panel) occurs for repulsive electron-electron interactions, which are slightly weaker for thick shells. This is a consequence of the larger electron delocalization, which translates into smaller |c 2 | coefficients in the CI expansion (see main text) and hence slightly smaller SU satellite, as observed in Fig. S5c.

Effect of inserting multiple impurities in CdSe/CdS
We consider here the possibility that two surface traps, instead of one, are acting as electrostatic impurities in CdSe/CdS NPLs. Since there is a general preference of forming defects in the heterostructure interfaces -because of lattice mismatch 10,12 -and on lateral facets -where ligand desorption is more likely to happen 6 -, we choose the charges to be located as shown in Fig. S6a. The presence of two charges, combined with the weak in-plane confinement, easily dissociates the trion by driving one electron to each surface impurity. This can be seen in the charge densities of Fig. S6b. The number of visible SU peaks, however, remains one (see Fig. S6c). In the case of strong surface charges (Q = 1.0), the trion triplet