Material parameters used in the work
The energy structure of multi‐layer core/shell semiconductor quantum dots (QDs) is simulated and the optical absorption is described within the linear response theory. The lattice‐mismatch strain field of the multi‐layer nanostructures of spherical symmetry is modelled by a linear continuum elasticity treatment. The excitonic effect is estimated by a configuration interaction approach. Multi‐layer core/shell QDs of II‐VI semiconductors with heterostructures of type I and II are discussed. Localization of the photo-excited carriers induced by coating explains the optical stability of these multi‐layer nanostructures.
- quantum dot
- optical absorption
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  or laser applications , and probably the most important in the immediate future, to molecular diagnostics and pathology . At theoretical level, the study of magnetism  or photon entanglement  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. ) has some limitations as the predictions are dependent of
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
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. ). 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, , has radial symmetry and consequently the field is
where and are relative lattice‐mismatches, 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 point of the valence band (VB) and conduction band (CB) for direct band gap semiconductors are given by the equation :
where the unstrained (bulk) values are related by (with , the unstrained (bulk) band gap), and is the volume deformation potential (subscript
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 . The one‐band effective Hamiltonian is
where is the photoexcited (electron or hole) carrier
where the upper sign corresponds to
where are constants and are spherical Bessel functions with the argument dependent of (). The functions
where and the prime are used to denote the first radial derivative. From the condition of normalization and the explicit form of functions
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
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 , 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  and of order 0.1 meV in CdSe/CdS QD with thick shell . The correlation interaction is also of order 1–2 meV or smaller . 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
Next, we limit discussion at modelling the excitonic effect by the Coulomb electron‐hole interaction mediated by a
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 are Coulomb matrix elements and is the homogenized screened relative dielectric constant. In the concrete calculus of the Coulomb matrix elements, we used the series expansion of the Coulomb Green function in spherical harmonics
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 . At the low‐light irradiation power, the assumption of a linear relation between the polarization and the electric field is a common approximation. By treating the QD‐field interaction as a perturbation, absorption coefficient expression can be obtained using either first‐order complex susceptibility or Fermi’s golden rule. Here, in the proof, we consider Fermi’s golden rule as follows. For a monochromatic electromagnetic wave , propagating in
Expression of the probability rate for absorption in the first order of approximation, , can be obtained following the standard textbook derivation of Fermi’s golden rule as
where is the optical matrix element between the initial and the final states, , and
Eq. (12) can be applied to obtain the excitonic absorption by computing the optical matrix element , where
, one obtains
where ; the last equality is obtained by making use of the slow spatial variation of the envelope wave functions over regions of the unit cell size and the orthonormality of the Bloch cell wave functions. By introducing the Kane momentum matrix element, , with and considering the polarization unit vector,
Then, expression of the single QD absorption coefficient from Eq. (12) becomes
where . The multi‐shell character is brought into the excitonic optical matrix element,
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
where is the concentration of QDs in solution and ln is the natural logarithm function. The model is based on a probabilistic hitting QD as a target and details of the derivation can be found in Ref. . For (that is QDs with radius smaller than approximately 10 nm) and (that is for dilute solutions), we obtain .
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).
|(eV)||2.25 e||2.69 e||1.74e||2.49e||3.61 e|
|(eV)||-5.34 e||-6.07 e||-6.00 e||-6.42 e||-6.6 e|
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. . To obtain the SPS structures, we take the version of Eqs. (5) and (6) for core/shell structure (see, for example, Ref. ). 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) is presented for a shell thickness of 4.3 nm. The SPSs energetically predicted location by the calculus of the eigenvalues is confirmed (checked) by the geometrical location. The degeneracy of the states is dictated by the angular momentum quantum number
For comparison, we make several remarks on the SPS structure we obtained (see Ref. ) by the
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. ). The PDs for the SPSs can be seen in Ref. . 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 (
In Figure 6 (reproduced from Ref. ), we show the energy value of FEA, , the single QD absorption coefficient for the corresponding FEA (obtained by Eq. (15)), and the colloidal absorption coefficient for the corresponding FEA (obtained by Eq. (17)) ; the argument
Regarding the location of the photoexcited charges, we introduce the radial probability density , where for electron and hole, respectively. One observes that since from the orthonormalization condition. The radius expectation value of the photoexcited electron or hole is obtained with . The results are remarkable by revealing the ZnS coating effect. Thus, coating is not significant for CSS QDI, the electron and hole remain located in the core not close to the core/shell interface. On the other hand, for CSS QDII, the hole radius expectation value is practically not affected by the ZnS coating and the hole remains close to the centre but the ZnS coating has a strong effect on the electron localization, namely, the electron moves to the middle of the ZnSe shell. Numerical details are given in (Ref. ). Thus, according to our modelling in both CSS QDI and CSS QDII, the electron and hole are not confined in the proximity of the interfaces and surface. Consequently, QD coating induces larger photoexcited carrier‐trap (interface defects and impurities close to the core/shell interface or to the outershell‐environment interface) separation. This might have an important effect regarding the competition between recombination and trapping. Thus, one of the interesting effects of the QD coating is the blinking suppression observed, for example, in ultra‐thick‐shell CdSe/CdS . According to our results, the model assuming a tunnelling barrier between the photoexcited carriers and the trap states, similar to that proposed in Ref.  can simplely explain the non‐blinking as follows. Coating induces increasing of the barrier thickness and consequently a lowering of the trapping probability. Then, under a continuous photoexcitation, (stimulated) recombination is more probable than the trapping and a continuous luminescence (non‐blinking) is observed from coated QDs. Given the low trapping probability heuristically concluded by our modelling, for the non‐bliking in coated QDs, we rather advocate the barrier model than the model of Auger recombination of the photoexcited carriers on the trapping states as proposed in Ref. .
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
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