Controlled Growth of Large SiO2 Shells onto Semiconductor Colloidal Nanocrystals: A Pathway Toward Photonic Integration

The growth of SiO2 shells on semiconductor nanocrystals is an established procedure and it is widely employed to provide dispersibility in polar solvents, and increased stability or biocompatibility. However, to exploit this shell to integrate photonic components on semiconductor nanocrystals, the growth procedure must be finely tunable and able to reach large particle sizes (around 100 nm or above). Here, we demonstrate that these goals are achievable through a design of experiment approach. Indeed, the use of a sequential full-factorial design allows us to carefully tune the growth of SiO2 shells to large values while maintaining a reduced size dispersion. Moreover, we show that the growth of a dielectric shell alone can be beneficial in terms of emission efficiency for the nanocrystal. We also demonstrate that, according to our modeling, the subsequent growth of two shells with increasing refractive index leads to an improved emission efficiency already at a reduced SiO2 sphere radius.


Modelling:
In this supplementary material we provide some details about the theoretical model used.We consider a quantum dot placed at the center of a dielectric sphere with refractive index .Assuming purely dipolar quantum dot emission, we model the radiative properties of the system by a radiation point-like dipole placed at the center of the dielectric sphere.The calculation involves the expansion of the electric radiated field by vector spherical harmonics.

Multipole expansion
In this section we briefly illustrate the multipole expansion of electric and magnetic fields.First, we remind that, for a monochromatic field with angular frequency , the real electric and magnetic fields are given by the Ansatz where is the dielectric constant of the sphere, are the vacuum magnetic permeability and dielectric permittivity, respectively.Then we can uncouple the electric field taking the double curl, obtaining  In turn, the radius-dependent scalar functions and satisfy the equations

where
. We note that these equations coincide with the spherical Bessel equation for ) , and are coefficients to be found by imposing the boundary conditions (BCs) for the continuity of tangent electric field (azimuthal and zenithal components), the radial component of the displacement vector and the full magnetic field at the dielectric sphere surface .Using Eqs.(1.3,1.4),one can define an electric multipole field of order by where In turn, using Eqs.(1.1,1.2) we obtain electric-like transverse magnetic (TM) fields (1.17) and magnetic-like transverse electric (TE) fields Hence, the vectorial spherical harmonics expansion of an arbitrary field with both TE and TM components is given by the superposition 3 (1) where in terms of the vector spherical harmonic with orthogonality properties ( ) Electric dipole in a sphere In this section we show how to find the coefficients of equation (1.19) in the case where we place a dipole in the center of a dielectric sphere of radius R. We then consider an oscillating electric dipole radiating monochromatic light in a sphere with relative dielectric permittivity .The real electric and magnetic fields over all space are in turn given by where is the Heaviside step function, and are the electric [magnetic] fields internal and external to the sphere, respectively.
By expanding the electromagnetic field in vectorial spherical harmonics one gets where is the relative dielectric permittivity of the medium places outside the sphere and is the electromagnetic field radiated by the dipole in the absence of the spherical boundary, given by the expression 1,4 By imposing the BCs for the continuity of the normal component of the displacement vector, tangential components of the electric field and continuity of the magnetic field we can calculate the coefficients and by solving the algebraic system from which we obtain We are only interested in because it is the amplitude of the far field radiated field by the system, which we will use to calculate the radiative efficiency, see next section.

III. Nanostructure radiative efficiency
In this section we show how to obtain the radiative properties of the system and how to derive the nanostructure radiative efficiency, which is the ratio of the radiated power by the dipole inside a sphere to the one radiated by a dipole in air.First, we remind that the time averaged Poynting vector in the outer region is given by

S8
For an electric dipole field in a uniform dielectric, we obtain In turn we can calculate the radiated power in the far-field region as Growth of a second spherical shell Finally, if we consider a dielectric shell coating the previous sphere the fields become and ( ) ( ) where 1  is the dielectric constant of the internal sphere,

2
 is the dielectric constant of the shell and 3  is the dielectric constant out (in our case air).

S10
Boundary conditions for the continuity of the normal component of the displacement vector, tangential component of the electric field and continuity of the magnetic field provide ( ) The time-averaged Poynting vector in the outer region is in turn given by The radiated power in the far-field region is in turn given by ( ) The nanostructure radiative efficiency, remembering that , is in turn given by ( ) Legendre polynomials that can be expressed by Rodrigues' turn, solutions are given by functions of first and second kind that can be expressed in term of the spherical Bessel function as the curl equations (1.3, 1.4) one can relate and to the magnetic and electric field respectively by a differential operator L obtaining1

Figure S1 :
Figure S1: Transmission electron microscopy characterization of the CdSe/CdS sample used as

Figure S2 :Figure S3 :
Figure S2: a-d) Representative TEM bright field image of a silica coated CdSe/CdS sample after

Figure S6 :Figure S7 :
Figure S6: Normalized optical absorption and PL emission of silica coated CdSe/CdS nanocrystals

Figure S8 :Figure S9 :
Figure S8: Representative TEM images of the surface of silica coated nanoparticles whose last

Figure S10 :
Figure S10: a) schematic of the set of DOE experiments performed for the second injection

Figure S11 :
Figure S11: Representative bright field TEM images of silica coated CdSe/CdS nanoparticles

Table S1 :
Level and corresponding concentrations of the eight samples of first DOE

Table S2 :
Level and corresponding concentrations of the eight samples of second DOE

Table S3 :
Level and corresponding ratios with NPs concentration of the eight samples of third Experiment