Efficient Modeling of Optical Excitations of Colloidal Core-Shell

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A: New Tools and Methods in Experiment and Theory

Efficient Modeling of Optical Excitations of Colloidal Core-Shell Semiconductor Quantum Dots by Using Symmetrized Orbitals Tiberius Ovidius Cheche, and Yia-Chung Chang J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.8b09758 • Publication Date (Web): 28 Nov 2018 Downloaded from http://pubs.acs.org on December 3, 2018

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The Journal of Physical Chemistry

Efficient Modeling of Optical Excitations of Colloidal Core-Shell Semiconductor Quantum Dots by Using Symmetrized Orbitals Tiberius O. Cheche,† Yia-Chung Chang*,‡,§ †University ‡Research §Department

of Bucharest, Faculty of Physics, Bucharest, Romania, EU

Center for Applied Sciences, Academia Sinica, Taipei, Taiwan

of Physics, National Cheng-Kung University, Tainan, Taiwan 70101

ACS Paragon Plus Environment

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ABSTRACT: An efficient method for the theoretical investigation of optical properties of semiconductor core-shell quantum dots (CSQDs) is introduced within the multi-band k  p approach, which takes the advantage of the symmetry of the system. The heteroepitaxial strain and excitonic effect are included in the calculation of energy levels, envelope wave-functions, exciton binding energy, and linear absorption coefficient. The adoption of symmetrized orbitals allows improvement of the computation time significantly. To avoid appearance of spurious solutions caused by imbalance of basis functions adopted, we consider an 8-band k  p model which is block-diagonalized into two conduction bands and six valence bands, that we call 2+6band model. The band non-parabolicity effect is modeled by an energy-dependent k  p term, such that the density of states obtained can mimic the actual density of states of a full-band model. The simulated absorption spectra of ZnTe/ZnSe CSQD are in good agreement with those observed experimentally, including the high rise of absorption at energies far above the absorption edge.


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I.

INTRODUCTION

Semiconductor quantum dots (QDs) are materials of considerable interest because of their novel optical and electronic properties. The size, shape confinement, and lattice-mismatch induced strain are factors used in tuning the physical properties of such nanostructures. Heteroepitaxial strain can cause significant changes of the energy bandgap when the shell is pseudomorphically grown on a small and compressible nanocrystalline core.1,2 Nanoscale colloidal core/shell quantum dots (CSQDs) of nanoscale shell thickness with reproducible and controllable size and shape with high quantum yields can be prepared with low cost by chemical synthesis.3,4 In this context, the nanostructures of spherical shape open new gates for both fundamental physics and technology. For example, spherical CSQDs are used in laser applications5 and studies of photons entanglement.6 The main challenge of the theoretical investigation of such mesoscopic structures is the efficient computation for a large number of atoms in the system. The role of the heteroepitaxial strain is considered by first-principles calculations, by using, for example, density-functional theory (DFT),7 a parameterized density-functional tight-binding method8 or a local density approximation.9 However, limitations of these ab-initio calculations (e.g., bandgap underestimation and high computational cost for larger number of atoms) make it difficult to apply them for the study of CSQDs with size larger than 5nm. Though there are studies on the size-dependence of light absorption in solutions of colloidal semiconductor quantum dots10,11, an efficient modeling which considers accurately both geometrical parameters (finite size and definite shape) and excitonic effect, according to our knowledge is still lacking in the case of mesoscopic semiconductor core-shell quantum dots of realistic sizes.


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In this context, to eliminate the mentioned existing drawbacks, we built a program whose main characteristics are as follows: (i) the problem of large number of atoms apparent in the atomistic calculations is overcome by use of the multi-band k  p approach in predicting the energy level structure; (ii) the strain effect due to geometrical restrictions are considered by an elastic continuum model; (iii) the excitonic effect is taken into account via a configuration interaction (CI) approach; (iv) the nanocrystals have zinc-blende structure and are of spherical shape. The framework in which our program is developed allows clear specification of its accuracy limits, as it will be shown. Related to characteristics (i) and (iv) it is worth mentioning that the spherical shape of the nanostructures does not alter the point-group symmetry of the associated k  p Hamiltonians for the CSQD. The zinc-blende materials lack inversion symmetry and have Td symmetry. Dresselhaus’s bulk inversion asymmetry (BIA) term describes the lack of inversion symmetry for zinc-blende materials.12 As the effect of BIA is typically small, we choose to neglect the BIA term in the present study, so the point-group symmetry of the system becomes Oh. Regarding characteristics (ii), in principle, an atomistic calculation of the strain field is more accurate than that by a continuum model for mesoscopic nanostructures. Yet, the continuum elasticity approach in the limits of homogeneous and isotropic materials has been shown to be in good agreement with the valence force field (VFF) models for semiconductor QDs of spherical shape and cubic symmetry (see, e.g., Pryor et al.13, Andreev et al.14). This fact and the dependence of the VFF results on the a priori information regarding the interface structure and surface passivation15 lead us to adopt a continuum model for the strain effect in CSQDs. More explicitly, we assume a defect-free multi-shell model, in which the strain field is generated only by the lattice mismatch. Such heteroepitaxial nanostructures with low concentrations of defects and impurities are obtainable in the laboratory.16


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Methodologically, the expansion of the envelope wave-functions in terms of plane-waves17-20 is one of the most used techniques within the k  p theory to predict the energy levels of the nanostructures. Basically, Schrödinger's equation is solved by using the Fourier transform to a superlattice formed by a periodic array of QDs. It is well known that these periodic boundary conditions introduce artificial physical interactions (electronic coupling and strain field deformation due to cell vicinity effect), which can be reduced by choosing appropriate size of the supercell.20 However, for charged states of CSQDs, the effects of artificial long-range Coulomb interaction introduced by the array would be difficult to remove. Thus, managing a single QD embedded in a matrix of definite shape and finite size could be a potential drawback for the plane wave method. To overcome this difficulty, we introduce a localized basis set to expand the envelope wave-functions assumed by the k  p treatment for the study of a single core/shell nanoparticle. In the multi-band k  p framework, there are two main approaches for dealing with the problem of the heterostructure boundary conditions. One of them uses a symmetrized Hamiltonian17 and the other replaces the material-dependent zone-center Bloch functions by a complete set of orthogonal periodic functions over the whole structure.21,22 In this work, we adopt the former approach, which generally is valid when the Luttinger parameters of the constituent materials of the heterostructure are not too different. In solving the eigenvalue problem, we adopt the weak solution method (see, e.g., Lassen et al.23) and integrate over the whole volume of CSQD to obtain the Hamiltonian matrix elements within the localized basis set. This method is a convenient alternative to the method of imposing continuity of the envelope wave-function and probability current at the core/shell interface to satisfy the boundary conditions. Related to the strain field it is important to mention that our method is applicable to core/multishell structures. Analytical expression of the strain tensor of such structures exists in


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the literature24 and it can be implemented in the electronic structure computation of the multiband k  p Hamiltonian including strain. In this work, we only consider core/shell nanostructures. The structure of the paper is as follows. In Section II we describe theoretical aspects related to basis states used in obtaining the single particle states (SPSs), symmetry considerations used to reduce the computational effort, the Hamiltonian model for the CSQD, computation of the absorption spectra by including the excitonic effect. Section III contains simulation results and comparisons with experimental data of the linear absorption coefficient for ZnTe QD and ZnTe/ZnSe CSQDs. Section IV encloses conclusions regarding the usefulness of the model we introduced and its associated numerical tool.

II. THEORETICAL MODEL Here we describe the basis states used in simulations and how to expedite the computation by taking advantage of the time-reversal and parity symmetry operation. Explanations regarding the Hamiltonian used, the boundary conditions for the CSQD heterostructure, and the selection rules of the optical transitions are presented. Finally, the excitonic effect is discussed.

II.1 LOCALIZED SPHERICAL BASIS STATES Next, we describe the localized basis set used to find the energy levels of CSQDs. Within the k  p approach the CSQD multi-band Hamiltonian is obtained by replacing the wave vector

operator k by  i in the bulk Hamiltonian. Solution to the time-independent Schrödinger equation is described in terms of products of zone-center Bloch functions and envelope wavefunctions. As a useful concept, 'the total angular momentum', F  J  L , where J is the Bloch angular momentum and L is the angular momentum associated to the envelope wave-function, is


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introduced by Sercel and Vahala.25 Considering zinc-blende semiconductors, they investigate spherical QDs embedded in a semiconductor matrix by neglecting strain. Spherical approximation (isotropic valence-band (VB) energy dispersion) for the kinetic Hamiltonian of Td double group symmetry leads to analytical solutions for the single particle states (SPSs). Generally, solutions for the semiconductor heterostructure QD Hamiltonian can be obtained efficiently by using basis sets that take into account both the crystal group symmetry and the shape (geometrical) symmetry of the QD. To take the advantage of the spherical shape of CSQD, we expand the envelope wave-functions within a complete set of eigenfunctions of a hard-sphere Hamiltonian. To avoid artificially strong quantum confinement due to the infinite barrier, we consider a hard sphere larger than the actual size of the CSQD. Thus, the localized orthonormal envelope wave-functions have the form CnL jL  knL r  YLm rˆ  , for r  R 0,for r  R 

 nLm (r )  r nLm  

,

(1)

where R  rcs  rb is the radius of the CSQD plus a buffer shell, rcs is the CSQD radius, rb is the thickness of the buffer shell, see Figure 1 ( r0 denotes the core radius). The buffer region refers to the range in which the electron can leak out and assumes that it terminates at R and the excitation energies are considered with respect to the ground state energy.


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Potential (eV)


ZnTe

ZnSe

core

shell

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buffer

4 2.25

r0

0

2.69

rcs

R

0.69 4

rb Figure 1. Scheme of the bulk band-edge alignment in unstrained ZnTe/ZnSe heterostructure. The numbers (in eV) show the lineup considered in simulation. It is found that using rb  0.3nm is sufficient for allowing the leak-out effect to manifest. Our calculation confirms that for larger values of rb the single particle state (SPS) energy structure remains unaltered in the accuracy limit of the model. In eq 1, L and m are the angular momentum quantum numbers, CnL2  2 R 3 jL 1  xnL  , jL is the spherical Bessel function, knL  xnL / R , xnL is 2

the n-th root of jL , YLm are the spherical harmonics, and rˆ denotes the polar angle of vector r that characterizes the carrier position. The spin-orbit coupled zone-center Bloch states are characterized by the angular momentum J. Consequently, the desired basis states for the product of envelope wave-functions and Bloch states are formed by the orthonormal eigenfunctions of the total angular momentum F, that we name as localized spherical basis states (LSBS) and write them as


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 FFz JLn  knL ; F , Fz ; J , L 

J

C

J z  J

F , Fz J , J z ; L, m  Fz  J z

nLm J , J z ,

(2)

where CJF,,JFzz; L,m  Fz  J z are the Clebsch-Gordan coefficients and J , J z zone-center

Bloch

functions.

 FFz JLn (r )  r knL ; F , Fz ; J , L   J J

z  J

We

use

are the spin-orbit coupled the

notation

C JF,,JFzz; L ,m  Fz  J z  nLm (r ) J , J z . To adapt to more general case

(possible to include inversion-breaking structures), we classify the eigenstates of the QD Hamiltonian according to the Td double group symmetry. The Oh irreducible representations have the same labels ( 6 , 7 , 8 ) as Td, (we adopt the notations from Koster et al.26) except that for Oh group the states with different parities (related to the inversion operation) are decoupled with labels ( 6 , 7 , 8 ). Once the BIA terms are included, the 6 and 7 states are coupled to   form 6 states, 6 and 7 states are coupled to form 7 states, while 8 and 8 states are

coupled to form 8 states for the Td group. The following useful symmetry considerations can be used to improve the calculation capability.

II.2 SYMMETRY OF LSBS FOR SPHERICAL QDS For spherical QDs with Oh symmetry, the symmetry types of SPSs can be identified by a projection of the eigenvectors obtained from using LSBS with definite parity and time-reversal character into states of definite symmetry for the Oh group. On another hand, the total angular momentum representation can be decomposed into the

6 , 7 , 8 irreducible representations of the Td group or in a similar manner into 6 , 7 , 8 irreducible representations of the Oh group (given the compatibility relations between Oh and Td group as expressed, e.g., in Table 88 of Koster et al.26). We can see that by decomposing the


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product of representations for the envelope wave-function nLm (which transform according to full rotation group L representations) and for the zone-center Bloch basis JJ z

(which

transform as the irreducible representations 6 of the conduction-band (CB) states, 7 ( J  1 / 2) , and 8 ( J  3 / 2) of the VB states). For example, for L  0, 1, 2,... one obtains rotational group

L 

nLm

Td irreducible representations

8   7   6 VB: J 3/2, J 1/2, J 1/2

 6   7   8 ;  2  2  4 ;  7 8  6      3 3 9 7 8;  6  5 6  5 7  98 ;

L0 L 1 L2 L 3,

(3)

In eq 3 we have used decomposition of the full rotation representations L for the even envelope wave-functions

under

Td

symmetry,

namely,

L  0  1 ,

 L1   4 ,

L2  3  5 ,

L3  2  4  5 . (see, e.g., Table 82 and 86 from Koster et al.26). Consequently, for both uncoated QD as well as coated QD with zinc-blende (diamond) symmetry, the Td (Oh) symmetry should manifest in the energy-level multiplicity. Thus, SPSs have maximum four-fold and minimum two-fold (Kramers) degeneracy, a consequence of the time reversal symmetry.

II.3 CSQD HAMILTONIAN Next, we describe the Hamiltonian of the CSQD, including the effect of strain,

HD  Hk  Hs,

(4)

where H k and H s are the kinetic and strain Hamiltonian, respectively, with cubic symmetry for the two constituent materials of the CSQD. As the piezoelectric effect is negligible for spherical structure27, we disregard this effect in the present study. The band-offset of the core/shell heterostructure is schematically represented in Figure 1. The zero energy reference is taken as the unstrained VB edge of ZnTe.


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Though the 8-band model widely used in the simulations works well for low-lying states of type-I CSQDs, we found that it can give spurious in-gap states when applied to type-II CSQDs. This is caused, within the LSBS adopted here, by the unbalanced weighting of the conduction and valence bands of the shell material. Although such a problem can be overcome by using two sets of LSBS’s (one for the core and the other for the shell), it makes the programming too tedious to be desirable. To avoid handling such complications, as mentioned, we consider a decoupled CB-VB Hamiltonian (2+6-band model), a good approximation for wide band-gap materials as considered here. In this 2+6-band model the CB reduces to the one-band effectivemass model. To simulate the band non-parabolicity effect and extend the validity of the k  p Hamiltonian to larger k around the  point (the extreme point for the direct bandgap semiconductors of the heterostructure considered). We introduce an isotropic energy dependent electron effective mass which is used to fit the density of states (DOS) of the full-band model obtained by DFT calculation. This is the main feature of the model which enables the simulation of the high rise of absorption spectra of CSQDs observed experimentally at energies far above the absorption edge. As for the VB dispersion relations of II-VI semiconductors, because they have less significant non-parabolicity in different crystal directions within the VB energy windows considered in our calculations, the Luttinger-Kohn Hamiltonian (excluding spin) takes the same form as the lower 3  3 block of the H k matrix28 (see Supporting Information, eq A1). Since the CB and VB are decoupled in this model (via a transformation), the  1, 2,3 modified parameters should be replaced by the original Luttinger parameters  1L, 2,3 . Including the electron spin and adding the spin-orbit interaction converts the model into a 6-band model for the VB states.


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To estimate the nonparabolic CB dispersion energy in the 2+6-band model we use the 4  4 Kane Hamiltonian18,29,30 (see Supporting Information-section A). One finds, for example, that along the [001] crystal direction the CB has the dispersion relation of the form E  Ec 

2 2  h2k 2 1  * *h k  EP  ,  c  E  Ev '    p ( E  Ec )   p c 2m0 E  Ev '  2m0 

E P  2m0 P02  2 ,

where

P0   ih s px x m0 ,

(5)

 p  1   1  4 2 ,

 c*   c  EP  Eg    1  2 (3Eg ) and  c  m0 me ; s px x is the Kane momentum matrix 1

element, Ev '  Ev   / 3 (  the spin-orbit splitting), Ev is the VB edge, Ec is the CB edge,

Eg  Ec  Ev is the semiconductor bandgap,  1 and  2 are the modified Luttinger parameters, and me is the electron effective mass. For the energy dependent electron effective mass, from eq 5 for E  Ec , we infer the following empirical isotropic form of the CB energy dispersion

E  Ec  

h2 k 2  c* ( E  Ev ' )   ( E  Ec )   ( E  Ec )2  EP 2m0 E  Ev ' h2 k 2 h2 k 2   c (E) 2mc ( E ) 2m0

,

(6)

where mc (E ) is the electron energy-dependent effective mass,  c (E ) is the energy-dependent CB Luttinger parameter, and α, β are parameters introduced for describing the nonparabolicity. Notice that in eq 6 β simulates the term proportional to  p c* in eq 5. At E  Ec , from eq 6 we have  c ( Ec )   c   c*  EP ( Eg   / 3) 1 or  c*   c  EP ( Eg   / 3)1 , a relation which is close to

 c* given in eq 5 for  / Eg = 1 . The analytical expression of DOS per unit volume (including spin degeneracy) related to the dispersion curve described by eq 6 is given by


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g(E) 

h2 k 2  2    k E   dk   2 1st Brillouin 2mc ( E )   1

zone

3/2

1  2m ( E )   E  Ec dmc ( E )   2  c 2  1   ( E  Ec ) 2  h   mc ( E ) dE  .

(7)

This expression is used to fit the DOS obtained by DFT calculation by adjusting the parameters α and β which enter eq 7 through eq 6. For the fitting procedure we consider an energy window of 4.2 eV above the conduction-band minimum. The energy-dependent effective mass, mc ( E ) defined in eq 6, with α and β obtained by fitting, is then used to obtain the CB eigenvalues of the CSQD. To find the self-consistent solutions to eq 7 for the CB states of CSQD, we first find all eigenvalues ( Ei ) of the energy-dependent effective mass model within the LSBS as functions of the input energy E in the range of interest. We then find the intercept of the family of curves described by Ei ( E ) with a straight line with slope 1. These intercept values correspond to the self-consistent solutions of eq 7. Detailed derivation of the kinetic, strain Hamiltonians are presented in Supporting Information-section A. To take into account the shape and size of the heterostructures in modeling the strain field we adopt the continuum elasticity theory for the strain of an inclusion in a finite elastic body.31 The isotropic continuum elastic strain tensor for spherical core/shell heterostructure from Pahomi and Cheche24 (see Supporting Information-section B) is implemented in the numerical code.

II.4 BOUNDARY CONDITIONS FOR THE HETEROSTRUCTURE Next, we discuss the boundary conditions problem. The Schrödinger equation is formulated as


13


N

N

1

1

Page 14 of 42

H D  A    E  A   ,

(8)





where   stands for one of the N basis states of the LSBS,  nFFz JL . Instead of imposing the continuity of the wave-functions and probability current at the interfaces, within the weak formulation we solve the eigenvalue equation N

H

,  ' 1

D  '

A'  EA ,

(9)

which results after applying a 'bra' LSBS wave-vector to eq 8 for a particular truncation of N. The Hamiltonian matrix elements (HMEs) that enter eq 9 are written as D D D H  k n ' L ' ; F ' F ' z ; J ' L'  '    H   '  k nL ; FFz ; JL H J

J'

 C

J z  J J 'z  J '

FFz JJ z ;Lm  Fz  J z

* D C JF' J'F'z'z;L 'm ' F 'z  J 'z  drnLm  Fz  J z (r ) H JJ z ; J ' J ' z n ' L 'm ' F ' z  J ' z (r ) 

,

(10)

where, as mentioned, the H JJD z ; J ' J ' z operators are obtained by replacing the k operator for the bulk Hamiltonian by the gradient operator  i , that is, H JJD z ; J ' J 'z  JJ z H D  i  J ' J ' z ;  is the volume of the CSQD plus the buffer shell. For the buffer region outside the CSQD, the band parameters are taken to be the same as those for the shell material except that confining potential barriers for the CB and VB states are taken to be Vc  0 and Vv  0 , respectively. These confining barriers simulate the band offsets between the shell material and the surrounding of the colloidal CSQDs. As a result of the Hamiltonian hermiticity, application of the weak method requires use of symmetrized operators H JJD z ; J ' J 'z (see, e.g. Baraff and Gershoni17) that are of the form f ( r )k k or f ( r )k  (not counting the terms without k dependence), where f ( r ) is a real function associated

to the piecewise-continuous Luttinger parameters (for the kinetic Hamiltonian) or to the strain


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tensor components (for the strain Hamiltonian) of the CSQD and  , are the Cartesian coordinates x, y, z . The symmetrization procedure implies the replacements

f k k 

1 1 k f k  k f k  , fk   k f  f k  .  2 2

(11)

The HMEs obtained from eq 11 involves integrals of the form

   f     dr *    f       dr * f     



  dr     f      f in *



 dr       f  *

out

in

out

dr *    

(12)

for quadratic in k operators and

 f    f in  dr*     f out in

 dr    *

(13)

out

for linear in k operators;  is the notation for the wave-vector nLm ,  in is the core volume, and out is the shell plus buffer volume. Thee integral in eq 11 involves Cartesian derivatives of the piecewise-continuous function f ( r )  f in   f out  f in  ( r  r0 ) , where  is the Heaviside function. The integrals from eq 10 are checked against a second method of calculus which consists in inserting the closure relation

 

   nLm nLm  1ˆ , n , L ,m

and the results are essentially the same within the desired accuracy. Complete details regarding the HMEs calculus are given in Supporting Information-section C.


15


Page 16 of 42

II.5 TIME-REVERSAL AND PARITY Due to the time-reversal symmetry, all SPS states are at least two-fold degenerate (Kramers degeneracy). Since the basis states k nL ; F , Fz ; J , L and k nL ; F , Fz ; J , L are related by timereversal, we only need to keep one half of the basis states with a selected Fz . The simplest way is

to

choose

Fz  1 / 2,3 / 2,..., (1) ( F 1 / 2 ) F (called

“Kramers

set

1”)

and

Fz  1 / 2, 3 / 2,...,( 1)( F  3/2) F (called “Kramers set 2”). The time-reversal operation relates the Clebsch-Gordan coefficients for two Kramers partners (with Fz and  Fz ) by the mapping

CJF,,JFzz;L,m  Fz  J z  ( 1) L J  F CJF,,JFzz;L, m  Fz  J z . It can be shown that the Hamiltonian matrix defined within the two Kramers sets of LSBSs are block-diagonalized and each set gives identical energy eigenvalues as those obtained with the full basis (see Supporting Information-section D). It is also convenient to classify LSBS according to their parities (with respect to inversion symmetry) since BIA is neglected here. Groups of LSBS with different parity are naturally decoupled due to inversion symmetry. The consideration of time reversal and parity immediately reduces the size of the Hamiltonian matrix by a factor 4 approximately. Such a consideration is applicable to all shapes of QDs with inversion symmetry, including cubes, ellipsoids, cylindrical, and rectangular rods.

The LSBS with odd parity can be constructed by the combination:

(Envelop wave-functions with even L)  VB Bloch states (p-like) +(Envelop wave-functions with odd L)  CB Bloch states (s-like). Similarly, the LSBS with even parity can be constructed by the combination: (Envelop wave-functions with even L)  CB Bloch states (s-like) +(Envelop wave-functions with odd L)  VB Bloch states (p-like). The odd parity of the electric dipole operator and the definite parity of the SPSs impose that an excitonic state is formed by pairs of electron-hole SPSs of opposite parity. By using the


16

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compatibility relations between Oh and Td group (see, e.g., Table 88 from Koster et al.26) we decompose the product of Td irreducible representations of the electron-hole SPSs as

( 6  7  8 )  ( 6  7  8 )

and

obtain

the

sum

of

irreducible

representations

21  3 2  53  7 4  85 (see, e.g., Table 82 from Koster et al.26). That is the exciton level multiplicity in a crystal with either Td or Oh symmetry is 1, 1, 2, 3, 3. Regarding the pairs of electron-hole SPSs, since the photon behaves like a p-state, which has 5 ( 4 ) symmetry under Td (Oh) group, we keep in the SPSs only those LSBSs which generate pairs of 5 ( 4 ) symmetry. For each manifold of the types 6  7 ( 6  6 , 7  7 ), 6  8 ( 6  8 , 7  8 ), and 7  8 ( 7  8 , 6  8 ) we only need to select the linear combination that

transforms according to one partner of the 5 ( 4 ) representation. The other resulting manifolds  6   7 ,  6  8  7  8 may be excluded as they are not optically active because of parity ,

reasons. For each manifold of the type 8  8 ( 8  8 ) (the pairs 8  8 and 8  8 are not optically active because of parity reasons), we only need to keep two linear combinations (due to two occurrences of 5 in the direct product) that transform according to the same partner of the

5 ( 4 ) representation (see, e.g., Table 82 from Koster et al.26). For manifolds of the types 6  6 and 7  7 , no excitonic states can contribute to the optical spectrum. They correspond to the “dark” excitonic states.

II.6 OPTICAL TRANSITIONS AND SELECTION RULES Next, we describe the optical transitions. For computational convenience we rewrite the λ- th SPS in terms of the Nb LSBS orbitals as


17




  (r ) 

n , F , Fz , L , J

where

An, F , Fz , L , J  nFFz JL (r ) 



j

D j (r ) j ,

Page 18 of 42

(14)

j is the abbreviated notation for JJ z and α is the abbreviation for (nLm). The

probability density (PD) is obtained by integration over the unit cell of the zone-center Bloch functions as

 (r )   |  D j r  |2 , j

(15)



that is, a weighted sum of squared envelope wave-functions. Optical matrix element between two i f SPSs,  i and  f is given by M if   eˆ  p  , where p is the electron linear momentum

operator and eˆ is the polarization unit vector of the photon. In the 2+6-band model (where CB and VB are decoupled), we may approximate (see, e.g., Gershoni at al.18)

    i e  p  f  e   Di ' j ' Dfj   ' p j ' j   j ' j  dR* ' ( R )  R  i  ', j '  , j   ,

(16)

where the capital R suggests integration over coarse-grained unit cells space (the envelope wave-functions space) and p jj '  j ' p j is the optical matrix element between Bloch states, which can be obtained by using P0 and EP (recall their definitions from eq 5). The first term in eq 16 characterizes the inter-band transitions ( j  j ' ), while the second is the intra-band contribution to transitions ( j  j ' ), which for UV-VIS spectra envisaged here may be neglected. The linear absorption coefficient of CSQD is calculated by using Fermi's golden rule, which, at low temperatures, is given by:24,32

 QD ( E , R) 

e 2  cn 0 m

2 0

Nv Nc

 M E v 1 c 1

 Evc ( R)  E  ,

2 vc

(17)


18

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where n is the refractive index, e is the electron charge, c is the speed of light in vacuum,

0

is

the vacuum permittivity,   4 R 3 / 3 , and Evc ( R )  Ec ( R )  Ev ( R ) . Nv and Nc denote the numbers of SPSs for VB and CB states included in the calculation. To simulate a solution of colloidal quantum dots, we assume that the QDs or CSQDs obey a Gaussian size distribution centered on the average radius R , with standard deviation  , P ( R, R,  ) 





 R R 1 2  1 1 exp   . 2 2  2  R  

(18)

Then, the average linear absorption coefficient is written as 

 QD ( E , R,  )   P( R, R,  ) QD ( E , R)dR 0



 9 1 e 2  Nv Nc 2  M vc  e 2  32  R cn 0 m0 v1 c1 0

R R12 2 2

 Evc ( R)  E  R 3 E ( R)

dR .

(19)

To solve the integral in eq 19, we empirically assume that the quantum-confined energy levels scale like R 2 with 2

R Evc ( R )  E g  [ Evc ( R )  E g ] 2 , R

(20a)

where for ZnTe QD, E g is the ZnTe energy gap. For CSQD we set Eg  qEc (core)  (1  q ) Ec (shell)  Ev (core) ,

(20b)

where q denotes the fraction of charge density inside the core for the CB ground state. Here we have assumed that the charge density in the top valence state is negligible outside the core. Thus, one obtains


19


 QD ( E , R,  ,  ) 

Page 20 of 42

9 1 e 2 128  R 3 cn 0 m02 E Nv

Nc

  M vc

2

v 1 c 1

2     E ( R )  E 1 1  vc g   exp  2 1      2 | E  Eg | Evc ( R)  E g     .

(21)

II.7 EXCITONIC EFFECT Next, we describe the excitonic effect. To evaluate it, as mentioned we use a CI method. In the second quantization, the excitonic states are expressed as

  v v1 cc1 Cvc cc cv g , N

N

(22) 

denotes the many-body ground state (filled valence band), and ci ( ci ) creation

where g

(annihilation) operator of the electronic SPS "i", the index v holds for the VB band and c holds for the CB band; the index c  [1, N c ] with Ec 1  Ec  0 , whereas the index v  [1, N v ] with

Ev 1  Ev  0 . Due to the low resolution of experimental absorption spectra of CSQDs that we intend to simulate in the present work, we only consider the effect of electron-hole Coulomb attraction on the absorption spectra (leaving the effects of the polarization energy and electronhole exchange interaction to a more refined description of the spectra). The CI method requires solving the secular equation of the form33

  E

vcv ' c '

c

 Ev  E   vv ' cc '  Vvcv ' c '  Cv ' c '  0 ,

(23)

where Vvcv'c'    drh dre  *v rh   *c re  veh (re  rh ) c' re   v' rh  ,

(24)




20

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with veh (re  rh ) being the electron-hole Coulomb interaction. In the evaluation of Coulomb interaction matrix elements, Vvcv'c' , we adopted the homogenous static approximation of Maxwell Garnett (MG)34 for describing the effective-medium dielectric function of the core-shell structure. The dielectric function in static or quasi-static regime of an object formed by inclusion of a material into another is, in a good approximation, homogeneous if the object is small compared to the light wavelength. Thus, we have veh (re  rh ) 

e2  a re  rh

and  a   s  c  2 s  2 f c ( c   s ) c  2 s  f c ( c   s ) , where f c  r0 R  1

3

is the filling

factor.  c and  s are the static dielectric functions of the core and shell, respectively. By using the series expansion re  rh

1



 4 

L

  2 L  1

L  0 m  L

1

* rL r L 1YLm (rê )YLm (rˆh )

and carrying out the angular integrals, we can reduce the Coulomb matrix elements of eq 24 to the form

Vvcv'c'  

4 e2

a



L

   2 L  1  L  0 m  L

1

R

0

R

rh2 drh  re2 dre rL r L 1 cc* ', Lm ( re )  vv ', Lm ( rh ) , 0

(25)

* (rê )c* re c' re  and  vv ', Lm ( re )   drˆhYLm (rˆh )v* rh v' rh  . Note that where cc* ', Lm (re )   drêYLm

* both charge density terms  cc ',Lm ( re ) and  vv ',Lm ( re ) are real functions in our case. The double

integrals in eq 25 can be carried out efficiently via the Simpson rule (with even grids) by taking advantage of the semi-separable form of the term rL r L1 , which allows the double integration to be evaluated as two single integrations via a recursive method (with significant saving in


21


Page 22 of 42

computation time). This is a key step for speeding up the computation of the Coulomb matrix, which is one of the most time-consuming part in the code. Including

the

excitonic

effect,

the

optical

matrix

g eˆ  P    v v1  c c1 Cvc M vc (eˆ ) , where M vc (eˆ )   v eˆ  p  c N

N

elements

become

is the SPS optical matrix

element for polarization vector eˆ and P  i p i is the momentum of electrons that fill the VB N

at T=0K. Consequently, expression of the QD absorption coefficient, starting again with eq 16 and considering a Gaussian size distribution of QDs or CSQDs at low temperatures, becomes

 QD ( E , R,  ,  ) 

9 1 e 2 128  R 3 cn 0 m02 E Nv

Nc

   Cvc M vc 

where

v 1 c 1

2

2   E ( R)  E '  , 1 1 g  exp  2   1     2  E  E g' E ( R)  E g'    

(26)

E denote the energies of the excitonic states, and E g' = E g  E X ; E X denotes the

exciton binding energy whereas E g is given by eq 20b.

III RESULTS AND DISCUSSIONS Next, we present and discuss the results of simulations obtained for ZnTe QD and ZnTe/ZnSe CSQD. ZnTe and ZnSe are wide direct bandgap II-VI semiconductors. ZnTe/ZnSe CSQD is a type II heterostructure, in which ZnTe (ZnSe) is under compressive (tensile) strain. The strain of the two components described, as mentioned, by a continuum model is isotropic and the elasticity parameters are specific to each component (see details in Supporting Informationsection B). In Table 1, we list the bulk parameters used in simulation. Since the leading input parameters for the electron effective mass and the Luttinger parameters for the constituent


22

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materials of the ZnTe/ZnSe CSQD as reported in the literature have a range of uncertainty, we adopt (for all our calculations) the following sets of parameters  1  3.96 ,  2  1.11 ,  3  1.47 , and mc m0  0.124 for ZnTe and  1  4.9 ,  2  0.9 ,  3  1.8 , and mc m0  0.13 for ZnSe. The values of these parameters are in the range reported in the literature (see, e.g. Lawaetz36); the confining potential barriers for the CB and VB states are taken to be Vc  4eV and

Vv  4eV , respectively as illustrated in Figure 1.

Table 1. Bulk material parameters Parameter

ZnTe

ZnSe

E g (eV)

2.25a

2.69a

E p (eV)

19.1b

19.1

 ( eV)

0.92b

0.43b

ac

-4.61d

-5.6e

av

-0.07d

-1.0e

bv

-2.21c

-1.14g

dv

____ f

-4.1g



7.4h

9.1h

lattice constant (nm)

0.6104h

0.5669h

Young modulus (1010 Nm-2)

4.17i

4.51i

Poisson ratio

0.363i

0.376i

a in35; b in36; c in37; d

in38; e in39; f shear strain cancels for spherical core; g in40; h in41; i in42.

In the continuous model the shear strain for spherical core is zero24; consequently, no shear deformation potential enters in the core strain Hamiltonian, (there is no entering for core dv in


23


Page 24 of 42

Table 1). Canceling of the shear strain in the spherical core justifies the widely accepted approximation of negligible piezoelectric potential for thin shells.27

a) ZnTe QD In the following discussion, we consider the simulation of the optical absorption coefficient for colloidal ZnTe QD. For comparison with the experiment of Fairclough at al.43, we take QD with average radius of 1.8nm. Using the strategy described in section II.3, we fit the ZnTe DOS calculated by DFT via the Wien2k code44 with eq 7 and the electron effective mass from eq 6 for energy in the range 0 : 4.2eV . Reasonable fit of DOS is obtained with   17 and   3.7eV 1 (see Supporting Information-section E). Before presenting the results for the optical absorption, we introduce the SPS structure of the QD obtained in the current model. Convergence within 1meV precision for SPS with energy eigenvalues in the energy window 0.7 : 3.3eV is obtained with the maximum quantum numbers L = 10 and n = 25. The SPS eigenvalues are two-fold (Kramers) and four-fold degenerate (dimension of 8 irreducible representation). Going into details, we plot probability densities of the first six SPSs, obtained with eq 15, in Section F of Supporting Information. Given the 2+6 band model, the envelope wave-function symmetry manifests by appearance of orbitals s, p, and d (spherical, dumb-bell, and butterfly shaped for hole states, respectively), whereas the cubic symmetry is present only in the shape of hole PDs). Each SPS and its time-reversal partner (Kramers degenerate pair) have identical shape. Both electron and hole ground states consist of mainly s-type orbitals, with a slight deviation from the spherical shape in the case of the holes. Thus, our simulation is in agreement with the conclusion of pseudopotential calculations45 regarding the s-like character of both electron and hole


24

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envelope wave-functions in the ground state of the QDs (similar situation is obtained for the ZnTe/ZnSe CSQD that we discuss later). The simulation for the absorption coefficients is shown in comparison with the experimental results of Fairclough et al.43, in Figure 2. For the CI calculation of excitonic states, we include all electron-hole product states with transition energy less than 5eV. The whole computation takes about 12 minutes (45 minutes) without (with) considering the excitonic effect, on a Linux server (running with 1 CPU). An inhomogeneous broadening (due to size fluctuation) as described by a Gaussian distribution with   0.07 was adopted to produce the broadened spectra, following eq 21 for the case without the excitonic effect and eq 26 with considering the excitonic effect. The simulated energy position and oscillator strength of the fundamental absorption peak (first peak) match well with the experiment. The oscillator strength, DOS, and the excitonic effect are factors responsible in the simulation of absorption intensity. In addition, the use of energy-dependent effective mass for the CB states (recall eq 6) results in a reasonable agreement with the experiment of the simulated absorption spectra at higher energies. The absorption coefficient falls in the range of the reported values ( 10 4 - 105 cm -1 ), and at higher energy becomes comparable to that for bulk II-VI semiconductors;46,47 this is because at higher energy, the DOS becomes comparable for nanostructures and bulk. In accordance with the general findings for the bulk case (see, for example, Sturge48), one finds that including the excitonic effect enhances the absorption coefficient for the low-lying transitions and causes a red shift with an amount equal to the exciton binding energy, E X . The Coulomb attraction between electron and hole results in lowering the transition energy and thus a red shift of the absorption spectra. Furthermore, it results in the increase of the overlap between electron and hole envelope wave-functions, which


25


Page 26 of 42

generates larger oscillator strength and a larger absorption coefficient. The inset in Figure 2 shows the absorption peaks for a narrower QD size distribution (   0.003 ) obtained with eq 26.

Figure 2. Linear absorption coefficient for ZnTe QD of radius r0  1.8nm . X (no X) denotes the simulation with (without) considering the excitonic effect and Exp denotes the experimental graph. In Figure 3, the linear absorption coefficients for ZnTe QDs of radius r0  2nm, 2.2nm and, for comparison, 1.8nm, are shown. For larger ZnTe QDs, the calculated spectra have a red shift due to smaller confinement energies accompanied by a weakening of the excitonic absorption features due to the reduced oscillator strength within type-II structures consistent with the experimental results reported in the literature.43,49 In addition, the expected behavior of the exciton binding energy, that is, its decreasing with the QD radius is obtained: E X  0.209eV for

r0  1.8nm , E X  0.191eV for r0  2nm , and E X  0.176eV for r0  2.2nm . On another hand, at higher energy, the absorption spectra of QDs of different radii become similar, which is in accord with our previous observation, that the QD DOS and, as a consequence, the absorption too, approach the bulk limit at higher energy.


26

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Figure 3. Linear absorption coefficient for ZnTe QDs of radii r0  1.8nm, 2nm, 2.2nm . X (no X) denotes the simulation with (without) considering the excitonic effect.

b)

ZnTe/ZnSe CSQD

In the following discussion, we consider the simulation of the optical absorption coefficients for ZnTe/ZnSe CSQD. For the band-offset at the ZnTe/ZnSe interface, a key parameter in simulation, we adopt the value of 2eV in absence of strain (which is in the range reported by Yu at al.50 or close to the value of 1.96eV reported by Lo et al.35). The values of the buffer barrier thickness and of the confining potential barriers are kept unchanged with respect to the above setting for the ZnTe QD. According to the continuum elasticity model under which the strain calculation is performed, we discuss CSQDs with shell thickness larger than 2 monolayers (MLs). For ZnSe, 1ML corresponds to a thickness of about 0.3nm.51 Similarly to ZnTe, we fit the ZnSe DOS calculated by using the Wien2k code44 with eq 7 and the electron effective mass from eq 6 for energy in the range 0 : 4.2eV . Reasonable fit of the DOS is obtained with   10 and   1.05eV 1 (see Supporting Information-section E).


27


Page 28 of 42

Before presenting the results for the optical absorption, we discuss the SPS energy levels structure obtained by simulation. The results obtained with the setting L  10 and n  21 for core radius r0  1.8nm and various shell thicknesses are shown in Figure 4. The convergence of SPSs energy is obtained with a precision of at least 1meV for the energy window

E

max v

 0.7eV, Ecmin  1.2eV  that is 0.7 : 3.2 eV . As one can see in Figure 4, the energy of the

hole ground state is weakly affected by the presence of the shell and practically insensitive to shell thickness for thicker shells. This is caused by the strong localization of the hole in the core. On another hand, the energy of the electron ground SPS decreases with increasing shell thickness as the electron becomes more localized in the shell. Another apparent feature in Figure 4 is the DOS increases with the energy for both electron and hole states. This happens for the electron, partially as a result of our adoption of the energy-dependent electron effective mass, whereas for the hole due to the warping of valence-band structure. As the SPS energy goes beyond the VB shoulder band offset, the DOS also increases due to larger volume available to hold the hole states. The heteroepitaxial strain effect is also presented. It is found that the core is compressed (the hydrostatic strain is negative) and the shell is dilated (the hydrostatic strain is positive). As a result of simulation presented in Figure 4, we obtain the lattice mismatch strain induces a decrease of the CSQD bandgap defined as the difference between the lowest CB SPS and highest VB SPS in the presence of strain. In Figure 4b the VB and CB edges (band offset) of the two materials forming the CSQD heterostructure when the strain is taken into account are shown. The values of the VB and CB extrema at the  point with the strain consideration are given by the relation:52

Ev ,c  Evb,c  av ,c hyd ,

(27)


28

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where the bulk values are related by Ecb  Evb  Eg with Eg the bulk bandgap, and  hyd is the hydrostatic strain (see Supporting Information-Section B). Note that the VB of strained ZnTe appears insensitive to the change of CSQD radius, since its deformation potential av for VB is very small (see Table 1).

Figure 4. SPS energy levels of ZnTe/ZnSe CSQD versus outer shell radius, rcs, with shell thickness of (2ML, 3ML,…,10ML), and the core radius r0=1.8nm: (a) without strain; (b) with strain. The dot (circle) symbols show the VB and CB edges of the strained ZnTe (ZnSe). In (b) are represented only the hole levels of energy higher than -0.426 eV. The 0 reference energy is the unstrained VB edge of ZnTe. To obtain more physical information, as in the case of QD, we plot the probability densities of low-lying SPS states obtained with eq 15 for ZnTe/ZnSe CSQD of r0  1.8nm and shell thickness of 10ML (see Supporting Information-section F). Generally, for one hand, the VB orbitals keep similar characteristics as those of ZnTe QD as the quantum confinement of lowlying hole orbitals due to the shell is not much different. For the higher excited hole states the mixing of wave-functions nLm of different values of L yields various PD shapes. Similar to the ZnTe QD case, the VB PDs also reveal the cubic symmetry. On another hand, with the increase


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of shell thickness the electron SPSs in CSQD show increasing probability for the electron being localized in the shell, especially for low-lying states. This is a typical behavior for the type II semiconductor heterostructures. In Figure 5 we present the radial probability densities,   ( r ) , for the three low-lying (highest-lying) CB (VB) levels for two different shell thicknesses. We have  2   ( r )  r 2  |  DnLm , j jnL  r  | , Lm , j

(28)

n

 where coefficients DnLm are defined in eq 14. ,j

The three CB levels are doubly degenerate (Kramer’s degeneracy) on top of the (2L+1)-fold degeneracy due to spherical symmetry, while the three VB levels are 4-fold degenerate (with 8 symmetry). The radial probability densities of degenerate SPSs are identical. One can see the electron is preponderantly located in the shell for large enough shell thickness.

Figure 5. Radial probability density   ( r ) for ZnTe/ZnSe CSQD with r0  1.8 nm and

rcs  2.4 nm (a), and rcs  4.8 nm (b) of the three lowest-lying (highest-lying) CB (VB) levels. The energies (in eV) of the corresponding levels and the L quantum number for each CB level are given in the legends. Strain is considered.


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The results for the optical absorption coefficients are shown in comparison with the experimental results43 in Figure 6. As in the case of QD, for the excitonic effect we consider in the calculation an energy window of 1.8 ~ 4eV . The numbers of levels (each is doubly degenerate) for VB and CB SPSs are Nv=183, 226, 273 and Nc=32, 38, 41, for rcs=2.4nm, 2.7nm, and 3nm, respectively. For the desired energy window and precision of 1meV, the computation time ranges from approximately 2 (for rcs=2.4 nm) to 16 hours (for rcs=3 nm) on a single-node processor, about 80%-90% of the time being spent for the calculation of excitonic effect. One of the physical results obtained by simulation is the red shift increasing of the absorption spectra with the shell thickness. Thus, the optical absorption spectra calculated without considering the excitonic effect recover the confinement-induced red shift obtained in the experimental absorption spectra of CSQDs. For low-lying transitions, the absorption coefficient slowly decreases as the shell thickness increases. This is caused by the decrease of the electron and hole envelope wave-functions overlap with the shell thickness increase. Another aspect refers to the influence of the excitonic effect on the spectra. Thus, we obtain a similar behavior as that reported for ZnTe QDs, that is, the excitonic effect enhances the absorption coefficient for the low-lying transitions and causes a red shift of the spectra. The binding energies of the exciton ground state obtained in our simulation are: E X  0.133eV for 2ML, E X  0.115eV for 3ML, and E X  0.102eV for 4ML, respectively. The binding energies respect the general rule of decreasing with the nanostructure size. As expected, given the continuum model adopted, the agreement with experiment is better for larger CSQD structures. Despite the over-all good agreement, there is some discrepancy between the theory and experiment. Though the insets of the Figures 2 and 6 show a group formed by the first few excited excitonic states which, by increasing the broadening (  in eq 21 or 26), may generate a blue shifted bump (as observed in


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the experimental absorption spectra), our simulation cannot accurately reproduce the line shape of experimental spectra near that bump. The physical origin of the observed blue shifted bump remains to be clarified.

Figure 6. Linear absorption coefficient for ZnTe/ZnSe CSQDs of radius r0  1.8nm and various shell thickness: 2MLs in (a), 3MLs in (b), and 4MLs in (c). In the simulation, the broadening width is   0.1, 0.15, 0.17 , respectively. X (no X) denotes the simulation with (without) considering the excitonic effect and Exp denotes the experimental graph. The insets show the linear absorption coefficient for broadening   0.003 . The experimental results are from Fairclough et al.43.


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Finally, we comment on some specific features of the optical absorption. The absorption coefficient obtained with eq 14 is isotropic with respect to the polarization of light, that is the spectra are the same for light polarization along all three x, y, z directions for both QD and CSQD. In addition, we find that the optical matrix elements of the intra-band transitions (see eq 16) is about two orders of magnitude lower than that of the inter-band transitions and that justifies its neglect (as mentioned). Regarding the shape of the spectra, larger value of the standard deviation  in eq 26 is associated to the inhomogenous broadening. Though not considered by our model, the possible phonon replica (that overlap at low resolution of measurement) are implicitly simulated by large enough  .

IV CONCLUSIONS AND REMARKS By using a localized basis set within 2+6-band k  p approach, we characterized the electronic structure, photo-excited carrier distribution and linear absorption spectra of spherical semiconductor QDs and CSQDs, including the excitonic effect. We also simulated absorption spectra with an 8-band model and obtained the CB PDs, which also present the cubic symmetry already signaled in the simulations with the 2+6-band model for the VB SPSs. The low energy spectra of ZnTe QD obtained for the same set of data are similar for the two multi-band models. As mentioned in Sec. I, when applied to type-II CSQDs the calculation based on an 8-band model might introduce spurious solutions (in-gap energy levels) for some n, L and energy window settings because of the imbalance between the number of LSBSs used for CB and VB states composing the SPSs. Since it is tedious to overcome this problem with LSBS, we preferred a 2+6-band model for this work. We exploited the spherical shape and the cubic symmetry of the ideal (defect-free) nanostructures to speed up the computation. Innovative localized basis set and


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continuum modeling of strain field of finite inclusions in finite elastic body are introduced. By selecting only the optically active combinations of electron-hole product states with appropriate symmetry the computation time is reduced with more than two orders of magnitude. The code is robust, its outputs respect all the group symmetry requirements and does not generate spurious solutions or in gap states. The obtained absorption coefficients of realistic-size CSQDs have values close to those reported in the experiment and their spectral features are in reasonable agreement with experimental data for colloidal ZnTe/ZnSe CSQDs, including the high rise of absorption for high-energy excitations (up to 4eV). By considering the polarization and the electron-hole exchange interaction, the fine structure of CSQDs may be obtained and thus the method may further be extended to simulate the electronic structures and optical properties of biexcitons in spherical nanostructures for possible applications in quantum-entanglement devices. In Supporting Information we provide details of the 8-band k  p Hamiltonian including the effect of strain, the continuum model of the strain field for CSQD, the derived matrix elements of the k  p Hamiltonian within LSBS, the matrix elements of Coulomb interaction for calculating excitonic states, symmetry considerations based on time-reversal and parity, the comparisons of DOS calculated within the present energy-dependent effective-mass model and DOS calculated by DFT, and charge density plots of various single-particles states in CSQD.



AUTHOR INFORMATION

Corresponding author *E-mail: [email protected] ORCID Yia-Chung Chang: 0000-0003-1851-4651


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

ACKNOWLEDGMENTS

We thank C. T. Liang for assistance in obtaining DOS of ZnTe and ZnSe with DFT. This work was supported in part by Ministry of Science and Technology, Taiwan under grant no. MOST 107-2112-M-001-032 .



REFERENCES

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