Wavefunction Engineering of Type-I/Type-II Excitons of CdSe/CdS Core-Shell Quantum Dots

Nanostructured semiconductors have the unique shape/size-dependent band gap tunability, which has various applications. The quantum confinement effect allows controlling the spatial distribution of the charge carriers in the core-shell quantum dots (QDs). Upon increasing shell thickness (e.g., from 0.25–3.25 nm) of core-shell QDs, the radial distribution function (RDF) of hole shifts towards the shell suggesting the confinement region switched from Type-I to Type-II excitons. As a result, there is a jump in the transition energy towards the higher side (blue shift). However, an intermediate state appeared as pseudo Type II excitons, in which holes are co-localized in the shell as well core whereas electrons are confined in core only, resulting in a dual absorption band (excitation energy), carried out by the analysis of the overlap percentage using the Hartree-Fock method. The findings are a close approximation to the experimental evidences. Thus, the understanding of the motion of e-h in core-shell QDs is essential for photovoltaic, LEDs, etc.

S3 energy in the range of 1.49 to 3.85 eV at different k-points, which is very similar to that obtained (2.7 eV) using our proposed model for pure CdSe QDs. The correlation value calculated by the DFT comes in the order of 10 -4 eV (0.07 meV) and the exchange value is in the order of 10 -2 eV (3.7 meV). These values can be ignored as the Coulomb interaction is the predominant factor, which contributes to the direct band gap energy. Therefore, the calculations for Coulombic interaction effects of the excitons were performed.
Further to verify the veracity of the model, regression analysis of the eigenvalue and the wavefunction of the charge carriers for ultra-thin shell and the pure core QDs were carried out. The Adj. R 2 value came nearly unity for both the electron and hole probability density function (PDF) showing nearly full recovery between the pure core and core-shell (ultra-thin shell) QDs. Both the parameters were analyzed using the multiple linear regression model. MUMPS (multi massively parallel sparse direct solver) has been employed to solve the Schrödinger's partial differential equation provided in COMSOL Multiphysics 10 software package. The QDs are solved in a one-dimensional (1-D) array, spaced equally at 0.005 nm each, this array contains equally spaced nodes which are considered as an electron or a hole, and have taken that there is one exciton per NCs. Using this, the overlap probability is determined, and the excitation energy is consequently calculated from the energy (eigenvalues) solving Schrödinger's equation.
The Schrödinger equation is solved using Hartree-Fock self-consistent mean field approximation due to the mean potential field applied by the charge carriers on each other.
Initially, the electron and hole are taken as non-interactive, as there are no Coulomb forces applied to the carriers, the Schrodinger's equation is solved for them individually in an iteration. Using Poisson's equations, the Coulomb interaction between the electron and hole is calculated from initial wave functions, which further modifies the confinement potential in the Schrödinger's equation at each cycle. 11 The self-consistent loop is controlled using COMSOL S4 live link with MATLAB 12 (Fig. S3). This programming is done to calculate the convergence between the initial and new wavefunctions and terminate the loop when convergence is attained. The electronic and optical properties, such as the binding energies, excitation energies, overlap integrals, overlap probabilities, oscillator strengths, the radial density of wave function, etc., are determined using the Origin analysis software package. The calculation of the overlap probability and the subsequent integration of the overlap probability with respect to the radius of the QDs is done on the Origin software. 13 Integration was carried out by package using the simple trapezoidal method. This approach is more accurate as the density of the data is high.

Electronic properties
The total energy of the single exciton is defined as 14,15 : Here, Eg is the band gap energy of the confinement region. The confinement region of the exciton is selected according to the relative integral of overlap probability of the exciton in the core-shell with respect to the QDs. In modeling, we have evaluated the perturbation by the Coulomb interaction, since the dimensions of the QDs semiconductor is within the Bohr's exciton radius for CdSe (RBohr = 5.6 nm). 16 The percentages of the overlap area of the coreshell regions were evaluated with respect to the QDs. The eigenvalues (εe, εh) were determined at the termination of the loop, εe−h is the attractive Coulomb energy, between the electron and hole. Since the attractive Coulomb energy is added twice (one is from εe and the other from εh), thus, one εe−h is subtracted from the total energy to rectify the error. The attractive Coulomb energy is determined using: 17 In Eq. 2, 0 and ℎ 0 are the single electron and single hole energies calculated in the first cycle taken without Columbic interactions. The exciton binding energy is defined as: 18,19 S5

Optical properties
For understanding the optical properties, a parameter that has to be evaluated is the overlap integral. 1,20 and is defined as: Here, Re(r) and ( ) are the radial wavefunctions of the electron and hole, respectively, solved from the self-consistent loops. To analyze the optical transitions, we require oscillator strength, which is a unitless quantity and defined as a transition density. 21 For the exciton, oscillator strength is given as: 17 Given here, and are defined as Bloch functions in the CB and VB, respectively, m0 is the free electron mass and E is the excitation energy. The Kane matrix elements can be simplified as: 22 where Ep is the Kane energy, 17.5 eV for CdSe, 19.6 eV for CdS. 23,24,25 From the equations (17) and (16), the electronic transition density is expressed as: And the radiative (natural) lifetime is defined as 26 : where, ε0 is the dielectric permittivity in vacuum, m0 is the free electron mass, c is the velocity of light, e is the electronic charge, f is the oscillator strength, n is the refractive index, E is the transition energy, and βs is screening factor given as: 27 Here, ε and are the optical dielectric constants of the medium and nanocrystal quantum dots, respectively. The confinement region of the exciton is determined by the overlap probability equation of the electron and hole as given by: 28 Overlap percentage is another such criteria to determine the confinement region of the electronhole overlap. Itis the relative coverage of the core/shell excitons with respect to the QDs exciton to distinguish the distribution of the excitons over the core-shell domain, which is given as 23 : Overlap percentage of the Type-I exciton = Here and are the radius of the CdSe and CdS QDs, respectively.  Figure S1. Overlap probability of the hole in the second sub-band of VB and electron of LUMO.   Figure S3. Algorithm for the self-consistent cycle of the mean approximation method for single exciton. S10