Effect of correlation and dielectric confinement on 1S1/2(e)nS3/2(h)Excitons in CdTe/CdSe and CdSe/CdTe Type­II quantum dots

We calculate correlated exciton states in type-II core/shell quantum dots (QDs) using a conﬁguration interaction method combined with the k · p theory. We map the 1 S ( e ) 1 / 2 1 S ( h ) 3 / 2 and 1 S ( e ) 1 / 2 2 S ( h ) 3 / 2 exciton correlation energy relative to the strong conﬁnement approximation as a function of core radius, shell thickness and dielectric conﬁnement. The type-II conﬁnement potentials enhance the e ↵ ect of dielectric conﬁnement which can signiﬁcantly a ↵ ect the wave functions and exciton energies in such heterostructures. Dielectric conﬁnement mainly increases the correlation energy for QDs in which the corresponding single-particle hole states are delocalized. We also ﬁnd that correlation leads to large changes in the optical dipole matrix element, particularly for the lowest CdSe/CdTe QD exciton, in the presence of dielectric conﬁnement. We conclude that dielectric conﬁnement af-fected the exciton properties in CdSe/CdTe QDs more than in CdTe/CdSe QDs due to the band alignment which encourages holes to localize in the shell. radiative excitons


I. INTRODUCTION
Semiconductor nanocrystals or quantum dots (QDs) are the subject of intensive research, due to a number of novel properties which make them attractive for both fundamental studies and technological applications. [1][2][3][4][5][6] QDs are of particular interest for solar cell applications due to their ability to increase e ciency via the generation of multiexcitons from a single photon. [7][8][9] QDs can be synthesized with a high degree of control using colloidal chemistry. 10,11 Much research e↵ort has been directed towards studying QDs grown from more than one semiconductor, e.g. core/shell heterostructures. [12][13][14] Such core/shell nanostructures provide a means to control the optical properties by tuning the electron-hole wave function overlap which is a↵ected by the alignment of the conduction band (CB) and valence band (VB) edges, as well as the QD shape and size. In contrast to type-I band alignments, type-II alignments have staggered CB and VB edges so the lowest energy states for electrons and holes lie in di↵erent spatial regions, leading to charge separation between the carriers. Type-II core/shell QDs can be classified according to whether the band alignments tend to localize the hole in the core and electron in the shell (h/e QDs, such as CdTe/CdSe QDs) or the electron in the core and the hole in the shell (e/h QDs, such as CdSe/CdTe QDs). 15 Such staggered band alignments have several useful physical consequences, including longer radiative recombination times for more efficient charge extraction in photovoltaic applications, 16,17 optical gaps that can be made smaller than the bulk values of the constituent materials 12,18,19 and control of the electron-hole wave function overlap which determines the exchange interaction energy. 20 Charge separation in type-II QDs can also be used to increase the repulsion between like-sign charges in biexciton states, 21,22 leading to the possibility of lasing in the single exciton regime. 6,23,24 To determine the energetics of many-body states in QDs, both the confinement potential and many-body interactions between the carriers need to be taken into account. Many-body interactions can be classified as Coulomb (charge) and Fermi (spin) correlation. Coulomb correlation arises from the electrostatic interaction of charge carriers in the many-body complex, whilst spin correlation occurs due to the fermionic character of the charge carriers (i.e. the Pauli exclusion principle). 25 Correlated many-body states may be calculated with the configuration interaction (CI) method which can be used in the framework of continuum or atomistic descriptions of single-particle states. [26][27][28][29][30][31] Colloidal QDs are usually embedded or dispersed in media 32 of lower dielectric constant than the semiconductor itself -this dielectric confinement leads to a modification of the Coulomb interaction which can be described using classical image charge theory. Whilst atomistic calculations 33 showed that dielectric confinement significantly a↵ects the charging energies of QDs, in singlematerial spherical QDs the similar electron and hole charge distributions lead to a weakened dielectric confinement e↵ect 34 on exciton states which mainly increases the binding energy. 26,35 It is therefore natural to wonder if the optical properties of spherical type-II core/shell QDs can be significantly a↵ected by the dielectric environment.
The e↵ect of dielectric confinement and many-electron correlation means that the single-particle picture is not good enough to faithfully predict exciton energetics or wave functions in colloidal QDs. The proper treatment of charges requires a many-electron description that goes beyond mean-field theories, and the configuration interaction method is one of the most accurate. However the full configuration method becomes progressively compu-tationally expensive as the number of states increases. Luckily however the interpretation of physical experiments often requires detailed knowledge of just a few excitons of particular symmetry. To overcome the unnecessary computational burden, when analysing 1S 3/2 , single-particle states into the exciton wave function but accurately reproduces the full CI results with greatly reduced computational cost. This allows us to evaluate the e↵ect of correlation on exciton energies and dipole matrix elements for many di↵erent core/shell QD geometries, and identify those designs for which correlation e↵ects are greatest.
In this paper we examine the 1S 3/2 excitons in CdTe/CdSe and CdSe/CdTe QDs using a CI approach to describe the e↵ect of correlation between the electron and hole states. The single-particle states were described using the (2,6)-band k · p theory 36 for spherical core/shell QD heterostructures, taking into account correct operator ordering at the heterointerfaces and the complex VB structure.

A. Single-particle states
In order to find a set of single-particle (SP) states we use the (2,6)-band k · p Hamiltonian. 36 To illustrate the problem and introduce basic parameters, in Fig. 1 we show: (a) a schematic of a spherical QD characterized by core radius a c and shell thickness a s , (b) the staggered alignments of the CB minimum (CBM) and VB maximum (VBM) in type-II h/e CdTe/CdSe and e/h CdSe/CdTe QDs; and (c) a characteristic profile of the self-polarization potential due to the contrast between dielectric constants of QD and surrounding medium (colloid). Details of material parameters used can be found in the Appendix.
The electron (hole) SP wave function where E e(h) j,p,n is the electron (hole) eigenenergy andĤ is the k· p Hamiltonian for electrons (holes). The SP quantum states are denoted using the notation, nl where n is the fundamental quantum number, l = s, p, d, ... represents the lowest value of the orbital angular momentum in the wave function, and µ 2 {e, h} denotes an electron or hole. 37 Due to the macroscopic spherical symmetry of the QD shape and the fact that the material parameters depend on the radial coordinate r only, the states calculated according the k · p theory can be characterized by the total angular momentum j and its z-component m ⌘ j z . 36 Furthermore the parity operator P commutes with the HamiltonianĤ of any system possessing a spherically symmetric confining potential V , so thatĤ andP share the same set of eigenfunctions. As a result SP eigenstates in spherical QDs are also characterized by the eigenvalue p of the parity operator; p takes the values 1 and 1 for even and odd states respectively. In spherical QDs possessing spherically symmetric confinement potentials the parity is conserved. Furthermore the radial part of the wave function can be classified according to whether it has odd or even parity. In Figs. 2 and 3 we show charge densities of n = 1, m = 1 2 electron and hole SP states with (a) j = 1 2 , (b) j = 3 2 and (c) j = 5 2 in CdTe/CdSe and CdSe/CdTe QDs respectively. SP states with j = 1 2 have spherically symmetric charge densities and all SP states with m = ± 1 2 are symmetric about the z-axis.

B. Exciton States
Our SP ket notation |jmpni is defined in terms of total angular momentum j and parity p, such that hr µ |jmpni = µ j,m,p,n (r µ ). To construct excitonic states we couple SP states in terms of angular momentum rather than parity, so we define the new ket notation |nljmi in terms of both total angular momentm j and lowest value of orbital angular momentum l. In such notation l = j p/2 for electrons and l = min(j + p/2, |j 3p/2|) for holes, where p = +1( 1) for even (odd) states regardless of whether electron or hole states are considered. 36 where V c is the interparticle Coulomb potential and V s is the self-polarization potential due to the interaction of a carrier with its own polarization charge. We note that p is the interface polarization potential. 15,38 Excitonic states are solutions of the Schrödinger equation where L is the total exciton angular momentum, L z is its z-component and E X is the exciton eigenenergy. The exciton wave function can be expanded in terms of uncorrelated electron-hole pair (EHP) states as 27 and the C L,Lz je,me;j h ,m h are Clebsch-Gordan coe cients. In Eq. (4) c is the expansion coe cient (character) of a particular ms where E X is the exciton energy calculated according to first order perturbation theory (FOPT) inside the strong confinement approximation (SCA) for the exciton wave function. 35,[38][39][40] The probability of excitation from the ground to the exciton state | L,Lz X i is proportional to the square of the optical dipole matrix element: 41 whereê is the polarization vector of incident light andp h is the hole momentum operator. Substituting for | L,Lz X i from Eq. (4) gives where each term in Eq. (8) must obey the selection rules for electric dipole transitions. Optical dipole matrix elements of the uncorrelated states are calculated as: To assess the e↵ect of correlation on the excitonic optical dipole matrix elements we define: 51 The charge density of the electron (hole) in the correlated exciton is However, since s (12) Similarly we define a hole RPD as where R µ;j,p,n J,l is the radial part of the electron (hole) wave function 26,36 and J is the total Bloch function angular momentum. The corresponding SP charge densities are denoted as ⇢ µ SP . We also define the probability p c(s) of the SP hole being in the core (shell) region as D. The e↵ect of dielectric confinement For colloidal QDs the dielectric constant " of the QD material is typically much larger than that of the surrounding medium. Such dielectric contrast means that any free charge in the QD induces polarization charge in the QD and its surroundings. The overall e↵ect of the induced charge on a source charge is described by the self-polarization potential V s (r) which cannot be ignored, Fig. 1(c). In colloidal core/shell QDs the self-polarization potential is characterised by a small peak and well near the core/shell interface due to the small dielectric mismatch between the core and shell materials. However a much larger peak just inside r = a c + a s and a deep well slightly outside the QD are due to the far greater dielectric mismatch of the shell and matrix material.
In order to assess the e↵ect of dielectric confinement on the excitonic structure of CdTe/CdSe and CdSe/CdTe core/shell QDs we performed CI calculations for two different situations: assuming a uniform dielectric constant " = const. = 6.65 (i.e. the mean of the CdTe and CdSe constants so that dielectric confinement by the surrounding medium and dielectric mismatch of the core and shell were neglected) and using a realistic profile " = "(r) with the individual dielectric constants for the core, shell and external medium. In the former case the Coulomb interaction V c in Eq. (2) reduces to the direct interparticle term V d only, allowing us to separate the e↵ects of the interparticle Coulomb attraction and dielectric confinement.

A. Convergence considerations
For a general exciton state | L,Lz X i there are many combinations of SP states that should be summed over in Eq. (5); this number can be reduced by considering the states that can be coupled for specific cases. Angular momentum coupling conditions mean that for optically active L = 1 states |j e j h |  1; assuming L z = 1 means m e + m h = 1. If incident light is polarized parallel to the z-axis only EHPs with m e = m h = 1 2 are excited. These assumptions still leave a large number of possible basis states to calculate matrix elements for, since |n e l e j e n h l h j h ; LL z i in Eq. (5) must be expanded over di↵erent ordinal quantum numbers n µ and angular momenta l µ . To investigate the relative importance of di↵erent SP states we calculate E X as a function of the number of states in the EHP basis. We include hole states up to j = 15 2 and all confined electron states in a full configuration interaction (FCI) scheme. Figure 4 shows the       21 . We see good quantitative agreement between the calculated exciton energies and the experimental data, with the data lying in the channels defined by an uncertainty of ±1 ML width in the core size. It should be noted that the results of Oron et al. 21 were obtained on zinc-blende NC structures, in addition to those of Cai et al. 44 The papers by Gong et al. 42 and Ma et al. 43 do not explicitly state the crystal structures of the core/shell nanoparticles, although Ma et al. 43 note that their core/shell NCs gave very similar absorption and photoluminescence spectra to those of Cai and coworkers. 44 Our calculations accurately reproduce the 0.25 eV energy separation between the 1S   15 In the CdTe/CdSe QD the electron is classified as shell-localized if its energy lies below the CBM of the core material, whilst the hole is core-localized if its energy lies above the VBM of the shell material. Similar criteria apply for the CdSe/CdTe QD. The type-I regime is defined by both carriers being delocalized over the entire QD, with their energies mainly determined by the 'global confinement' provided by the potential well of radius a c +a s . The type-II regime corresponds to the carriers being localized in di↵erent regions of the QD (i.e. the hole in the core and the electron in the shell for the h/e QD and the reverse for the e/h QD) so the SP energies are mainly determined by the dimensions of the relevant region. The quasi-type-II regime corresponds to partial charge separation; in the h/e QD this corresponds to a delocalized hole and shell-localized electron or a core-localized hole and delocalized electron in the e/h QD.
Areas of significant magnitude correlation energy in Fig. 6 always roughly coincide with those associated with large optical dipole matrix elements for the corresponding uncorrelated EHPs. 39 In those regions the electronhole correlation is enhanced as the SP wave function overlap is high and the interparticle Coulomb matrix elements are increased in magnitude. In the type-II regimes |E corr | is mainly small since the spatial separation of the electron and hole (induced by the type-II band alignment) overrides Coulomb attraction. In those regions the exciton wave function is closer to being described by the SCA. Although areas of high |E corr | partly overlap with the type-II regions the trend is for E corr ! 0 in the type-II localization limit. The highest |E corr | values in Fig. 6 are a consequence of dielectric confinement affecting the correlated hole density [see (a,iv) and (b,iv)], reflected by the fact that they mainly occur in regions where the corresponding SP hole is delocalized. For example, in CdTe/CdSe QDs dielectric mismatch increases |E corr | for the 1S    excitons in CdSe/CdTe QDs which lie below or near the hole LB (corresponding to delocalized SP holes). Figure  6(a, iv) and (b, iv) also show that E corr has a distinct minimum as a function of a s . This minimum is due to the fact that an increase in a s causes the hole density to shift into the shell where it starts to be a↵ected by the self-polarization potential at the QD/medium interface. However, as a s increases further the hole localizes completely in the shell so that spatial confinement by the VBM overrides the repulsive e↵ect of the self-polarization potential causing E corr ! 0; see Sec. III D 2 for detailed explanations.
For the 1S (e) 1/2 1S (h) 3/2 exciton the area of non-zero E corr in the lower right quasi-type-II regime of the CdTe/CdSe QD, Fig. 6(a,ii)), is equivalent to the area in the upper left quasi-type-II regime of the CdSe/CdTe QD, Fig.  6(a,iv). Similarly the region of large |E corr | for the 1S  Fig. 6(b,iv).
Overall we find that dielectric confinement a↵ects the correlated hole density more than the correlated electron density for two reasons. Firstly, the larger e↵ective mass and deeper potential well experienced by SP hole states compared to electron states allows the former to localize more fully in the shell, closer to the peak in V s (r) at r ' a c +a s . Secondly, the smaller energy spacing between the hole SP basis states (i.e. the larger density of hole states) compared to electron SP basis states means that the resulting correlated hole density has more 'degrees of freedom' to adjust to the e↵ects of dielectric confinement.  Fig. 6). We see that in the presence of dielectric confinement |E corr | . 20 meV for both excitons and that E corr exhibits at least one minimum as a function of a c in the " = const. and " = "(r) case. |E corr | is up to four times greater in the presence of dielectric confinement (" = "(r)) compared to the " = const. case. This result highlights the importance of a proper treatment of the dielectric environment in such nano-structures. The minimum in E corr for the 1S Fig. 7(a), is a consequence of two competing e↵ects: proximity of the self-polarization potential peak which tends to reduce the electron-hole separation and the e↵ect of the type-II confinement profile which tends to separate the carriers as a c increases. However, for the 1S 3/2 exciton E corr is not a monotonic function of a c either for " = "(r) or " = const. Insight into the QD size dependence of E corr can be gained by considering the amount of probability density in the core and shell associated with the dominant SP hole state that the correlated exciton originates from. The inset in Fig. 7(b) shows the amount of 2s (h) 3/2 hole probability density in the core as a function of a c , demonstrating a similar qualitative a c dependence to E corr in the dielectric mismatch case. We expect an increase in p c to cause a decrease in |E corr | since a greater amount of hole density in the core leads to less overlap with electron density in the shell and less correlation. 1S when (i) " = const. and (ii) " = "(r). The shifts due to the direct interparticle Coulomb interaction are relatively small when " = const. and both carriers move slightly towards the core. By increasing a c to ⇠1 nm and beyond, the hole mainly localizes in the core so that ⇢ h X ' ⇢ h SP and the SCA regime is reached, where the spatial confinement outweighs e↵ect of correlations.
When " = "(r) the increased values of |E corr | are associated with a shift of RPD away from the QD surface due to repulsion by the large peak in the self-polarization potential near r = a c + a s . The correlated electron is affected more since the electron SP states tend to localize in the shell. The exciton wave function gains 2s

CdSe/CdTe QD: e↵ect of hole shell localization
In Fig. 9 we present E corr for the 1S  Fig. 6). We observe the largest size correlation energy of the considered excitons for the 1S  3/2 state in CdSe/CdTe QDs when " = "(r), with E corr reaching 62 meV for an a c = 3.5 nm, a s = 0.6 nm QD, Fig. 9(a). This value is more than six times larger than the corresponding value for the " = const. case, highlighting the particularly strong e↵ect of the dielectric environment on this exciton. It can be observed from Fig. 9 that the ef- fect of dielectric mismatch on E corr is strongest in the vicinity of the 1s (h) 3/2 LB, i.e. once the SP hole becomes delocalized over the QD.
We found the second largest |E corr | value for the CdSe/CdTe QD 1S 3/2 LBs where those two hole states become delocalised over the QD. The maximum value of |E corr | is almost nine times larger in the presence of dielectric confinement compared to its absence. We note that the curves for E corr and the amount of hole RPD in the shell p s have qualitatively similar a s -dependences to the right of the 1s LB, see inset in Fig. 9(b). An increase in p s is associated with a decrease in the amount of hole RPD that overlaps with the core-localized electron leading to a decrease of electron-hole correlation and causing E corr ! 0. 1S 3/2 exciton: In the case of a core-only CdSe QD and no dielectric confinement, correlation causes both carriers to move towards the centre of the QD compared to their SP counterparts, Fig. 10(a,i). This is purely a result of the direct interparticle Coulomb interaction, giving correlation energies of approximately 5 meV. Introduction of the self-polarization potential, i.e. " = "(r), further exaggerates this move of RPDs of both carriers away from the QD surface in the core-only CdSe QD, Fig. 10(a,ii). This e↵ect increases localisation of both To assess the e↵ect of dielectric confinement on the correlated carriers in the CdSe QD we consider the expectation value of the 1s electron (hole) radial coordinate, denoted hr µ i. When " = const. (no self-polarization) we find hr h i = 1.55 nm compared to hr h i = 1.44 nm when " = "(r) for the 1s RPD has significantly greater overlap with the repulsive peak in self-polarization potential near the QD surface than the SP 1s (h) 3/2 hole RPD, the correlated electron is shifted by dielectric confinement by almost the same distance as the correlated hole. These results reflect the larger sensitivity of the correlated hole wave function to the dielectric environment compared to the electron in the CdSe core-only QD.
In Fig. 10(b,i) we see that the introduction of a thin CdTe shell allows the uncorrelated hole to start to localize nearer the QD surface (at r = a c + a s ), dramatically reducing its overlap with the uncorrelated electron. However when " = const. the introduction of correlation, even in the case of no self-polarisation, is strong enough to pull the hole back towards the centre, mainly due to the addition of the 1s  3/2 ( = 2) EHP to the exciton wave function. We see that the introduction of dielectric confinement again exaggerates this move of the carriers further away from the QD surface compared to the " = const. case. The e↵ect of dielectric confinement is particularly strong in this case because hr h i for the 1s (h) 3/2 state is close to the value of QD's outermost radius, a c + a s . The close proximity of the hole to QD surface reduces the distance ⇠ = hr QD h i hr image h i between the hole in the QD and its mirror image in the colloid, dramatically increasing the Coulombic repulsion between them which scales as / 1/⇠. Such repulsion causes the hole to be pushed back towards the centre of the QD, thereby dramatically increasing overlap with the correlated electron wave function. The presence of dielectric confinement means the exciton wave function is an almost equal superposition of the 1s (e) 1/2 ns (h) 3/2 (n = 1, 2) states, with |c 1 | 2 = 0.449 and |c 2 | 2 = 0.458 (inset in Fig. 10(b,ii)). For comparison, when " = const. the 1s 3/2 character amounts to only |c 2 | 2 = 0.019. The much stronger configuration mixing in the dielectric confinement case allows E corr to reach ⇠ 62 meV, compared to -9 meV without dielectric confinement.
Further increase of the CdSe/CdTe QD shell thickness to a s = 1 nm allows the SP hole to fully localize in the shell while the SP electron stays in the core, reaching the type-II localisation limit. The carriers e↵ectively enter the strong confinement regime in which the Coulomb e↵ects are overridden by the e↵ects of the type-II spatial confinement. In the SCA ⇢ SP , and the effect of correlations is lost. Again, E corr is only non-zero when the hole is delocalized; once it localizes in the shell, the e↵ect of VBM confinement overrides the interparticle Coulomb attraction. Dielectric confinement only slightly shifts the hole density towards the core, resisted by the opposing e↵ect of spatial confinement with E corr ⇡ 3 meV for a s > 1.5 nm. 1S 3/2 exciton: In the absence of dielectric mismatch (" = const.) the interparticle Coulomb interaction mainly causes the hole to move towards the core whilst the electron is nearly una↵ected, Fig. 11(a-c,i). Introducing dielectric confinement systematically moves the hole RPD towards the centre of the QD, whilst the electrons are again minimally a↵ected, Fig. 11(a-c,ii). Dielectric confinement has the most pronounced e↵ect on the a s = 0.5 nm QD for which the self-polarization potential is able to almost completely repel the hole RPD from the shell to the core region, Fig. 11(b,ii); this is associated with an increase of |E corr | to 36 meV. The movement of the correlated hole RPD to the centre of the QD upon introduction of dielectric confinement is due to the mixing of 1s (e) 1/2 ns (h) 3/2 (n = 1, 2) EHPs into the exciton wave function. For the a s = 1 nm QD the competing e↵ects of type-II VBM profile, interparticle Coulomb attraction and self-polarization potential lead to the correlated hole RPD being delocalized across the whole QD with a local maximum in both the core and shell. When a s is increased to 1.5 nm and beyond, the SCA regime is reached and the correlated RPDs are very close to those of the uncorrelated states (not shown).
Generally we have observed when the correlated wave functions are in the type-II localization regime the charge that localizes in the QD's shell is a↵ected more by the interparticle Coulomb attraction, whilst the innermost confined charge carrier is barely a↵ected. We explain this behavior from the fact that s Gauss's Law means that hole charge density ⇢ h X at some radius r h is not a↵ected any more by electron charge density ⇢ e X situated at r > r h .

E. Exciton optical dipole matrix elements
The regions of largest | P 2 X | in Fig. 12 closely coincide with the regions of largest |E corr | in Fig. 6 since the greater the change in carrier density due to correlation the greater the change in electron-hole wave function overlap. However, the correlation can increase or decrease the dipole matrix element of a particular exciton state relative to the SCA depending on the localization regime of the uncorrelated charge carriers.
Correlation only slightly changes the dipole matrix elements of the 1S closely follows the 1s (h) 3/2 LB (see Fig. 12(a,i)), indicating that the 1s (h) 3/2 hole should be delocalized (or approximately so) for correlation to reduce the dipole matrix element relative to the uncorrelated EHP. Once the hole localises in the core P 2 X becomes positive. Similar behaviour is seen when " = "(r), except that P 2 X can be slightly negative for QDs with thin shells, a s . 0.5 nm. Figures 12(a,iii-iv) show a similar trend for the CdSe/CdTe QD, with P 2 X only being negative when the 1s (h) 3/2 hole is delocalized; this is particularly noticeable for CdSe/CdTe QDs with a c & 1.5 nm when " = "(r) in Fig. 12(a,iv). As the shell width increases for a particular core size P 2 X increases dramatically near the 1s (h) 3/2 LB, due to the interparticle Coulomb interaction that prevents the hole wave function localizing in the shell and dramatically increases its overlap with the electron wave function in the core, Fig. 10(b). This e↵ect is enhanced by dielectric confinement, shown by the larger brighter area in Fig. 12(a,iv) above the hole LB. The di↵erence P 2 X reaches a maximum value of 0.15 for an a c = 3.5 nm, a s = 0.75 nm QD when " = "(r), compared to a maximum value of 0.042 (for an a c = 3 nm, a s = 0.625 nm QD) when " = const.. These shifts represent an increase by a factor of 7.29 and 1.5 in the dipole matrix element respectively relative to the SCA results found using FOPT. Both regions of positive P 2 X seen in Fig. 12 (a,iii-iv) near the hole LB are mainly due to the mixing of the 1s 3/2 LB in both the CdTe/CdSe and CdSe/CdTe QDs. This is most clearly seen for the CdSe/CdTe QD in Fig. 12(b,iiiiv), for which the reduction in the contribution of the 1s (e) 1/2 2s (h) 3/2 EHP leads to the an area of negative P 2 X . These areas roughly coincide with the regions of positive P 2 X in Fig. 12(a,iii-iv), suggesting a transfer of oscillator strength from the 1S = 13.4 ns) is about one order of magnitude larger than that calculated with DCI (⌧ DCI rad. = 2.85 ns). This reflects the need of proper treatment of correlation e↵ects in the design of optoelectronic devices that rely on dynamic process between charges.
We also found that the e↵ect of correlation on the CdSe/CdTe QDs exciton dipoles and radiative lifetimes strongly depends on the dielectric properties of the external medium. In Table I we list radiative times calculated with and without correlation for several values of " 3 . The first two rows correspond to the spatially varying dielectric constant " = "(r) whilst the third row corresponds to " = const., i.e. the case without dielectric confinement. Generally as " 3 is increased and the dielectric mismatch between the QD and the external medium falls we see that the e↵ect of correlation on the radiative lifetimes becomes less important. However for the case of a QD in vacuo or air (" 3 = 1) and commonly used solvents like toluene or hexane (" 3 ⇡ 2), the e↵ect of correlations on the radiative lifetime cannot be neglected. Type-II QD heterostructures have smaller dipole matrix elements than core-only QDs of the same ground state exciton energy. Such reduced dipole matrix elements could be beneficial for solar cell applications due to the longer radiative recombination times (and better charge extraction e ciencies) of photoexcited charges compared to core-only QDs. 45 3/2 exciton. The shaded area shows the region of as in which correlation has the greatest e↵ect.

IV. CONCLUSIONS
We developed a computationally e cient decoupled CI scheme to examine the correlation energy E corr and the change in optical dipole matrix element P 2 X of the 1S (e) 1/2 nS (h) 3/2 (n = 1, 2) excitons as a function of core radius and shell width in type-II CdTe/CdSe and CdSe/CdTe core/shell QDs. We have found: (i) The QD designs which gave the largest magnitude E corr values for the 1S 3/2 EHPs in the excitonic wave function caused by dielectric mismatch. (ii) The dielectric confinement mainly a↵ected QDs in the type-I and quasi-type-II localization regimes, particularly those QDs for which the corresponding SP hole states are delocalized. (iii) Overall CdSe/CdTe QDs were a↵ected more by dielectric environment than CdTe/CdSe QDs, as they tend to localize holes in the shell closer to the repulsive peak in the self-polarization potential that arises from dielectric mismatch. We conclude that the correlated holes are more a↵ected by dielectric confinement than the electrons due to the much larger density of states in the VB. (iv) The regions of (a c , a s )-space with the largest P 2 X corresponded to regions in which |E corr | was greatest. The dipole matrix elements of the 1S 3/2 (n = 1, 2) excitons can be significantly changed by dielectric environment in CdSe/CdTe QDs, in contrast to CdTe/CdSe QDs in which they are only slightly a↵ected. These results suggest that changing the dielectric environment could be another way in which to control the oscillator strength and radiative lifetime of excitons in CdSe/CdTe core/shell QDs.
In contrast to epitaxially grown QDs, which largely correspond to " = const. case in our analysis, our results show that the charge separation due to the type-II band alignments and by dielectric mismatch induces selfpolarization in core/shell colloidal QDs leads to strong deviations from the SCA for the exciton wave function.
The SP and excitonic states were calculated using the parameters in Table II where E g denotes the bulk band gap, the Kane energy is E P0 = 2m 0 P 2 0 /~2, P 0 is the bulk interband momentum matrix element, m 0 is the free electron mass, 41 is the spin-orbit splitting and E v is the VBM energy. The Luttinger parameters are L i (i = 1, 2, 3), ↵ is a CB parameter and m ⇤ e represents the electron e↵ective mass at the bottom of the CB, 37 " is the dielectric constant given in the units of free space permitivity, " 0 . Modified Luttinger parameters for the VB were calculated in the spherical approximation. 36