## 1. Introduction

Semiconductor quantum dots (QDs) are nanometre‐scale objects that are different to the usual optical materials, have size‐tuneable and narrow fluorescence and broad absorption spectra. The large interest for these materials comes from their technical applications and theoretical openings. For example, the colloidal multi‐shell QDs have led to the development of high‐efficiency solar cells [1] or laser applications [2], and probably the most important in the immediate future, to molecular diagnostics and pathology [3]. At theoretical level, the study of magnetism [4] or photon entanglement [5] is an example in which the theoretical studies emerge in promising new properties of electronic devices for spintronics or computer microchips. The main factors used in tuning the physical properties of QDs are the shape and size confinement, and the lattice‐mismatch‐induced strain. ‘Giant’ core/multi‐shell of 18–19 monolayers (MLs) shell thickness can be prepared with low cost by chemical synthesis [6, 7]. The theoretical predictions of electronic structures and optical properties are important in the core/shell quantum dots (CSQDs) engineering. Modelling of the lattice‐mismatch strain is a key factor in obtaining an accurate physical description of CSQDs. Widely used for analysing the linear elasticity of epitaxial‐strained heterointerfaces, the valence force field method (see, e.g. Ref. [8]) has some limitations as the predictions are dependent of *a priori* information regarding the interface structure and surface passivation. There are several theoretical studies of multi‐component nanocrystals, in which the role of the strain is considered by *ab‐initio* calculations [9–11]. Structural limitations of these first‐principle calculations (e.g. band gap underestimation) and also more important, the computational cost for larger QDs, make difficult comparison between theoretical predictions and the experiment. 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 models for semiconductor QDs of spherical shape and cubic symmetry (see, e.g. Ref. [12]).

In this context, in this chapter, we discuss optical properties of CSQDs. In Section 2, we introduce the theoretical modelling. In Section 2.1, we describe the lattice‐mismatch strain field in core/multi‐shell nanostructures by a continuum elasticity approach. In Sections 2.2 and 2.3, we describe the electronic structures and obtain the single‐particle states (SPSs) by an effective two‐band model and by an eight‐band model within the **k·p** theory, respectively. In Section 2.4, we discuss the excitonic effect in QDs. In Section 2.5, the optical absorption coefficient is obtained by taking into account the excitonic effect. In Section 3, we apply the theory presented in the previous section to several types of semiconductor CSQDs with heterostructures of type I and II. Conclusions are presented in Section 4.

## 2. Theoretical modelling of core/shell semiconductor quantum dots

The theoretical models introduced in this section combine knowledge from classical mechanics, solid state physics and quantum optics for description of elasticity, electronic structure and optical spectra in CSQDs.

### 2.1. Strain field and the band lineup in the presence of the strain

In the elasticity theory, description of the strain field in finite‐domain elastic bodies is a difficult problem. Elaborate solutions for such a problem are obtained in few particular cases for Eshelby‐type inclusion of a finite elastic body (see Ref. [13]). In what follows, we introduce an efficient simple method to obtain the strain field in concentric spherical domains of different elastic parameters, which satisfies the necessary accuracy level for our problem. We start by observing that for an elastic spherical core/shell structure, the displacement field, *irrotational*. Within the continuum elasticity approach, the equilibrium equation is grad *div* **u** = 0 [14]. To obtain the strain field, the linear stress‐strain tensor (*R* (for the total radius of core + middle shell + outer shell), we consider the following boundary conditions: (a) continuous stress at the interfaces, (b) zero pressure outside nanostructure and (c) *shrink‐fit* induced by the lattice‐mismatch. Transposed in algebraic equations, we have

where *A*, *B* and *C* denote the core, middle and outer shell, respectively. In Eq. (1c), which makes connection between the continuum elastic and the discrete crystalline approaches, *X* (*X = A*, *B*, *C*) should respect the above‐mentioned equilibrium equation, that in spherical coordinates **u**_{r}^{X}*r*^{2} (with *E* is the Young modulus and

The band offset is an important parameter in modelling the physical properties of semiconductor heterostructures. Simple relations are obtained for the band lineup in the presence of the strain for a two‐band model within the effective mass approximation. Thus, the energy values at the

where the unstrained (bulk) values are related by *v*(*c*) stands for VB(CB)). Within **k·p** theory, in a multi‐band approach, the kinetic and strain Hamiltonians are obtained according to the point group the semiconductors belong. By using a perturbation theory or a group theory‐based method (method of invariants), the expressions of the kinetic and strain Hamiltonians are written for the desired number of bands. For example, based on the work of Kane [19] and Bir and Pikus [20], expressions of the eight‐band kinetic and strain bulk Hamiltonians, including the spin‐orbit interaction for zinc blende crystals, are obtained in Ref. [21]. The multi‐band kinetic plus strain Hamiltonians are adapted to limited size structures by adding a cut‐off potential (corresponding to a buffer shell), which allows the electron to leak more or less into the environment, outside the structure.

### 2.2. Single‐particle states by two‐band model within the effective mass approach

The electron and hole SPSs are obtained by solving the Schrödinger equation for the envelope wave function

where *r*‐dependent effective mass,

where the upper sign corresponds to *E > V _{i}* and the lower to

*E < V*. Solutions of Eq. (7) are spherical Bessel functions

_{i}(5) |

where *f* are chosen according to the resulting band lineup (with taken into account the strain). Imposing the physical conditions of continuity, *any* relative position of the band lineup for the three regions:

(6) |

where *f*, we find *f* are given in Ref. [15].

### 2.3. Single‐particle states by multi‐band model within the k·p approach

The expansion of the envelope functions in terms of plane waves (see, e.g. Refs. [22, 23]) is one of the most used techniques within the **k**·**p** theory to obtain the energy structure of the nanostructures. By this method, Schrödinger’s equation is solved by using Fourier transform of a superlattice formed by periodic boxes containing the nanostructure that is analysed. To avoid the artefacts induced by the artificial periodic boundary conditions imposed by the array of boxes, which can affect the accuracy of the results especially when Coulomb and/or strain field interactions are present, we consider a single nanostructure whose Hamiltonian is solved in a localized basis set defined over the volume of the nanostructure. Regarding the inner (core/shell interface) boundary conditions, we adopt the *weak* solution method (see, e.g. Lassen et al. [24]) and integrate over the whole volume of nanostructure to obtain matrix elements between basis states (formed by the products of envelope functions and zone‐centre Bloch functions) for the Hamiltonian. This method is a convenient alternative to the method of imposing continuity of the envelope function and probability current at the core/shell interface to satisfy the boundary conditions. Details of computing the SPS energy structure by an eight‐band model for CSQDs are given elsewhere [25].

### 2.4. Excitonic effect

To obtain an accurate description of the optical properties, in addition to the geometrical confinement and strain, the excitonic effect should be considered. In order of their contribution to the excitonic effect, the interactions that are commonly considered are as follows: the Coulomb attractive interaction of the photoexcited electron‐hole pair, the polarization interaction of the charges induced at the interfaces, the electron‐hole exchange interactions and the correlation interactions [26, 27]. The polarization interaction, according to calculus of Brus [28], is about 3–5 times lower than the absolute value of Coulomb interaction and is smaller than the nanostructure size for ZnO, CdS, GaAs and InSb. The electron‐hole exchange interaction is smaller. For example, in spherical InAs QDs of radius 3 nm, it is of 2.093 meV comparatively to the Coulomb interaction of 60.6 meV [26] and of order 0.1 meV in CdSe/CdS QD with thick shell [27]. The correlation interaction is also of order 1–2 meV or smaller [29]. For the level of accuracy of the two‐band model introduced in Section 2.1, the Coulomb electron‐hole interaction yields a reliable estimation of the excitonic effect. The muti‐band **k**·**p** method is more accurate and, within this approach, calculus of all the above‐mentioned interactions is possible at the desired accuracy.

Next, we limit discussion at modelling the excitonic effect by the Coulomb electron‐hole interaction mediated by a *homogenized* dielectric constant (see below). We write the exciton state *α* as a configuration interaction expansion,

where the first and second term stand for electron and hole kinetic energy, and the third for electron‐hole Coulomb interaction, respectively. According to Maxwell Garnett (MG) formalism [31, 32] for spherical structures, the non‐local dielectric constant may be replaced by a homogenized value, and will adopt this approach in our calculus. With the algebra of the second quantization, one obtains the secular equation:

where

and the angular integrals separately factorize for electron and hole and they are computed analytically by using Gaunt’s formula.

### 2.5. Optical absorption

In this section, we derive an expression of the absorption coefficient, which is valid in the limit of low‐light irradiation power. We approximate the absorptive material as being formed by QDs in contact and assume that each QD is an absorber of volume *z* direction in a macroscopic non‐conducting media, the gradient of the energy density is (see, e.g. Ref. [33]) *single* QD that is,

Expression of the probability rate for absorption in the first order of approximation,

where *δ* is the Dirac function. To mimic the homogenous broadening that takes into account the relaxation phenomena of the carriers, one replaces the Dirac function by a Lorentzian of width

Eq. (12) can be applied to obtain the excitonic absorption by computing the optical matrix element **e** is the polarization light unit vector. Thus, by writing the transition linear momentum operator in the basis set

where **e**, parallel to the quantization axis, *z*, for example, one obtains

Then, expression of the single QD absorption coefficient from Eq. (12) becomes

where

by the domain dependence of the electron and hole single SPSs. The quantity expressed by Eq. (15) is the exciton oscillator strength. For the exciton state at resonance, one obtains that the absorption coefficient is proportional to the exciton oscillator strength. As the parameters entering Eq. (15) characterizes a single QD, we name it *single* QD absorption coefficient. Neglecting light scattering, a good approximation for QD of small size comparatively to the light wavelength, we introduce a *colloidal* absorption coefficient for the measured absorption coefficient in experiment:

where

## 3. Application of type I and II semiconductor heterostructures

In this section, we apply the theory developed in Section 2 to heterostructures of type I and II of direct band gap semiconductors. In accordance with our continuum model for strain, we consider QD structures with thicker shells. On the other hand, as a two‐band model neglects both VB‐CB and VB sub‐band mixing, we assess its validity in our modelling. As a first requirement in neglecting VB‐CB mixing, the semiconductors should be of wide enough band gap. Regarding neglecting of the VB sub‐band mixing, things are more complicated, and to assess to what extent a two‐band model respects a specific accuracy, a more advanced theoretical treatment is necessary. For this purpose, we compare the SPSs generated by the two‐ and eight‐band approaches for CSQDs of wide band gap semiconductors. For the modelling, we consider ZnTe/ZnSe CSQD (a type II heterostructure), ZnTe/ZnSe/ZnS CSS QD (a type II ZnS‐coated heterostructure) and CdSe/CdS/ZnS CSS QD (a type I ZnS‐coated heterostructure).

### 3.1. Comparison of two‐ and eight‐band approaches for wide band gap CSQDs

Before focusing discussion on quantum mechanics calculation of SPSs, we apply our method of computing the strain in lattice‐mismatch heterostructures. The results obtained by solving Eqs. (1) for the general case of different elastic parameters presented in **Figure 1** for CdSe/11CdS/ZnS (CSS QDI) and ZnTe/16ZnSe/ZnS (CSS QDII); the numbers before the chemical symbols hold for the number of MLs and the core radii are of 2 nm and 2.2 nm, respectively.

The parameters we use are listed in **Table 1**.

Once the hydrostatic strain and the bulk band offset are known, we can obtain the band lineup in the presence of the strain, by using Eq. (2). The bulk energy structure of the heterostructures (band offset) we discuss is schematically shown in **Figure 2** for CdSe/CdS/ZnS and ZnTe/ZnSe/ZnS (in what follows we take as zero reference the bulk VB edge of the core semiconductor).

ZnTe | ZnSe | CdSe | CdS | ZnS | |
---|---|---|---|---|---|

a (Å) | 6.08^{a} | 5.65^{a} | 6.05^{b} | 5.82^{b} | 5.40^{c} |

E (10^{10} N m^{-2}) | 4.17^{a} | 4.51^{a} | 2.87^{d} | 3.26^{d} | 5.55^{c} |

0.363^{a} | 0.376^{a} | 0.408^{d} | 0.410^{d} | 0.384^{c} | |

2.25 ^{e} | 2.69 ^{e} | 1.74^{e} | 2.49^{e} | 3.61 ^{e} | |

-5.34 ^{e} | -6.07 ^{e} | -6.00 ^{e} | -6.42 ^{e} | -6.6 ^{e} | |

0.79^{f} | 1.65^{f} | 0.9^{b} | 0.4^{b} | 2.31^{f} | |

-5.83^{f} | -4.17^{f} | -2.00^{b} | -2.54^{b} | -4.09^{f} | |

3.74^{g} | 3.77^{g} | 3.33^{h} | 4.11^{d} | 2.54^{g} | |

1.07^{g} | 1.24^{g} | 1.11^{h} | 0.77^{d} | 0.75^{g} | |

1.64^{g} | 1.67^{g} | 1.11^{h} | 1.53^{d} | 1.09^{g} | |

^{i} | 0.152 | 0.148 | 0.18 | 0.15 | 0.225 |

^{i} | 1.092 | 1.292 | 0.90 | 0.60 | 1.582 |

0.20^{j} | 0.21^{j} | 0.15^{k} | 0.22^{k} | 0.34^{l} | |

^{m} | 7.4 | 9.1 | 10 | 8.9 | 9 |

^{a}Ref. [35].

^{b}Ref. [36].

^{c}Ref. [37].

^{d}Ref. [38].

^{e}Ref. [39].

^{f}Ref. [40].

^{g}Ref. [41].

^{h}Ref. [42].

^{i}Calculated with

^{j}Ref. [44].

^{k}Ref. [45].

^{l}Ref. [46].

^{m}Ref. [47].

For 1 ML of ZnSe, we considered the thickness of 0.33 nm [48], and for 1 ML of ZnS, we approximated it as 0.33 nm also.

Next, we discuss the SPSs obtained for ZnTe/ZnSe CSQDs by the two‐band model introduced in Section 2. After calculus of the hydrostatic strain (by the method we mentioned above), we calculate the band lineup in the presence of the strain. For this, the bulk band offset assumed by Eq. (2) is take from Ref. [39]. To obtain the SPS structures, we take the version of Eqs. (5) and (6) for core/shell structure (see, for example, Ref. [16]). The eigenvalues of several electron and hole SPSs are presented in **Figure 3**. For a core radius of 2.2 nm, beginning with shell thicker than 3.1 nm, the lowest electron states start to be located in the shell while the highest hole states are practically insensitive to the shell thickness and located in the core. Our calculus shows that the first two or three (for shell thinner than 3.1 nm) hole SPSs are of heavy hole type.

To get a more complete image, in **Figure 4**, the probability density (PD) *L* and the PD shape by the values of *n*, *L* and |*m*| (which results from the

For comparison, we make several remarks on the SPS structure we obtained (see Ref. [25]) by the **k**·**p** method. According to the irreducible representations of T_{d} point group, the maximum value of degeneracy is four. Absent in the two‐band model, the time‐reversal symmetry is present in the eight‐band model, and all SPSs are at least two‐fold degenerate (Kramers degeneracy). The cubic symmetry of the zinc blende structures can also be observed in the PD shape. The shape of the PD, except the mentioned cubic symmetry, is similar in the two approaches. To have a more complete image of the density of states in the two approaches, we collect and compare the eigenvalue for CSQDs of different shell thicknesses. One obtains that the density of states is larger in the eight‐band model, and for these wide band semiconductors, this is mainly caused by the VB sub‐band mixing. However, when the shell thickness increase, the CB SPSs become denser and the discrepancy decreases. Consistently, in both approaches, one obtains the red shift of the SPS fundamental inter‐band transition with the shell thickness. The conclusion of the comparison is that if only several excited states are sufficient to be taken into account in modelling, then CSQDs with thicker shells and larger cores can be approximately characterized by a two‐band model. Generally, the VB SPSs are a mixture of heavy, light and split‐off hole states. The two‐band calculus shows, as expected, that the close to the VB edge, the SPSs are heavy hole state type, and this is another approximation that can be used in a two‐band modelling of this type of CSQDs. On another hand, in case higher excited states are necessary in modelling, then a two‐band model becomes inadequate. We can compare our two‐band model prediction with the experimental measurement of the fundamental absorption for this type of CSQDs with shell thickness of 5 ML. Thus, for core radius of 2.2 nm and total radius of 3.85 nm (corresponding approximately to 5ML ZnSe shell thickness), we obtain a transition of 517 nm (2.397 eV) comparatively to approximately 550 nm (2.254 eV) reported in Ref. [49]. Energy overestimation by our prediction is mainly due to neglecting of the excitonic effect. If we estimate the excitonic effect (binding energy) by an attractive interaction of 0.13 eV, then our prediction becomes 547 nm, close to the experiment. In the next section, we obtain an estimation of the binding energy from the calculus of the Coulomb electron‐hole attraction.

### 3.2. Excitonic effect and optical absorption

In this section, we evaluate the excitonic effect for CSS QDI and CSS QDII with thick outer shell. First, following the model from the previous section, by using Eqs. (1), (2), (5) and (6), we obtain the SPS structure. In the calculations, for the both nanostructure types, we find that the first four VB SPS states are heavy hole states and the fifth is the first light hole. Since the density of states is higher for heavy hole states, according to discussion from the previous section, we can appreciate that a two‐level approach is at least satisfactory if only the first several hole levels are taken into account in the modelling. The characteristics of the SPS structure are as follows. The red shift of the fundamental inter‐band transition with the middle shell thickness is found. The optical stability effect of the outer shell as reported by experiment [7, 49] is explained by the slight variation of the SPS energy with the ZnS shell thickness. Both electron and hole are located in the core for CSS QDI while for CSS QDII, the hole is located in the core and the electron in the shell, revealing the type I and type II character of the heterostructures, respectively. These observations are shown in **Figure 5** (reproduced from Ref. [15]). The PDs for the SPSs can be seen in Ref. [15]. With the SPS structure obtained, we estimate the exciton binding energy (or in an equivalent formulation, the fundamental excitonic absorption (FEA)) by computing the Coulomb electron‐hole attractive interaction by a configuration interaction method as introduced in Section 2.4. With the first four energy levels we consider, we form for both modelled nanostructures the set of configurations as follows. The SPSs have the quantum numbers (*n*, *L*) = (1, 0), (2, 1), (3, 2) and (4, 3) for electron and (4, 0) for hole, and each state is 2*L*+1 degenerated. Then, we build the configurations by replacing one from the set of (1 + 3 + 5 + 1) VB states with one of the (1 + 3 + 5 + 7) CB states, and obtain a total of 160 configurations.

In **Figure 6** (reproduced from Ref. [15]), we show the energy value of FEA, *x* represents the number of ZnS‐coating MLs. For the Kane energy, which is a necessary ingredient, we estimate **Figure 6(a)**, first, we setup the screened dielectric constant that is used to obtain the Coulomb matrix elements. Thus, we fit the experimental *E*_{FEA} of 620 nm from experiment [7] for CdSe/19CdS (CSQD with core radius of 2 nm and CdS shell thickness of 19 ML). Then, we analyse the effect of the ZnS coating and obtain the following results. *E*_{FEA} is slightly blue shifted with ZnS thickness and asymptotically blue shifted with the ZnS overcoating. Thus, a relative weak change of *E*_{FEA} with the ZnS overcoating as reported by the experiment [7] is obtained. This behaviour is primarily the result of the lattice‐mismatch strain and secondly of the excitonic effect. *α _{sol}*(

*x*), is weakly changed by ZnS coating for

**Figure 6(b)**as follows. First, to set‐up the screened dielectric constant, we fit the experimental

*E*

_{FEA}for ZnTe/6ZnSe (CSQD with core radius of 2.2 nm and ZnSe shell thickness of 6ML) of 570 nm from experiment [49]. Then, we analyse the effect of the ZnS coating and obtain the following results.

Regarding the location of the photoexcited charges, we introduce the radial probability density

## 4. Conclusions

We described optical properties of CSQDs of type I and II with thick shells by a two‐band model, which first is justified by a comparison with an eight‐band **k·p** model. To obtain the lattice‐mismatch strain effect on the energy structure, we developed a continuum elasticity model for isotropic, homogeneous and finite size multi‐layer structures. When implemented in the specific quantum mechanics of the multi‐shell semiconductor nanocrystals, the strain model we introduce is able to predict the main characteristics of fundamental absorption in thick shell CSQDs. By taking into account the excitonic effect, we can explain the optical stability of the overcoated core/shell QDs. According to our estimations, the measured absorption coefficient in colloidal QD solutions is practically not sensitive to the overcoating. The most important finding regarding the coating effect is related to the photoexcited carrier localization. According to our predictions, the photoexcited carriers are moved away from the surface and interfaces by overcoating. We believe that this work is an argument for improving the modelling. An eight‐band treatment able to describe the optical absorption of ZnTe/ZnSe CSQDs which considers high excited states is already done (see Ref. [25]). In this context, the analysis of the multi‐exciton generation, electron‐phonon interaction and relaxation dynamics are promising successful directions of research in finding responses to the complex optical processes that take place in multi‐layer QDs.