**4. Theoretical prediction of the electronic properties of heterostructed QDs**  The electronic band structure of the heterostructures (with/without alloy composition) quantum dots (QDs) was evaluated, using a k\*p model, considering specific features, related to the discontinuity of the effective mass, crystal potential and dielectic constant at the core/shell, as well as at the shell/surrounding interface. The evaluation also considered anisotropy of effective masses, particularly in the IV-VI semiconductors. The evaluations explored interesting properties, associated with charge distribution between the core and the shell, effective electron-hole spatial separation, probability of transitions, coulomb interactions and tunability of band-edge and remote states' energy.23, 48 Pre-engineering of the electronic band structure is done by theoretical consideration (of spherical particles alone for the moment), covering all cases, when, either the core or shell or both has alloy composition (see scheme in Figure 13). A few special points should be considered: anisotropy in effective mass (typical for IV-VI semiconductors), as well as the fact that each physical parameter dependents on its position (*r*) across the dot, and also may vary smoothly at the core/shell and the shell/surrounding interfaces, with a smoothing factor . The Hamiltonian was adjusted to the discontinuity at the PbSe/PbS interface by the appropriate choice of the kinetic energy term, ensuring probability current conservation and continuity of the envelope functions. In addition, the Hamiltonian potential energy term, included the heterostructure band offset, abrupt for the c/sh structure, but considered as a smooth function for the c/a-sh QDs.49 Presumably, the smooth potential profile reflects the nature of the interface region in alloyed materials with gradient composition, being an extension of the standard treatment for semiconductor heterostructures. The overall band offset was chosen as that of the corresponding bulk PbSe and PbS materials (where the valence band maximum of bulk PbS lies 0.025 eV above that of PbSe, while the conduction band minimum lies 0.155 eV above that of PbSe). Diagonalization of the envelope function Hamiltonian yielded the electron and hole wavefunctions, as well as a good approximation of the energy values of the conduction and valence-band's states. The heterostructures

*0* versus the detection energy across the PL spectrum, of 2.6

**1.15 1.20 1.25**

Detection Energy [eV]

**PbS 3.5nm CQDs**

 **4K 75K 140K 260K**

Fig. 12. Plots of the measured

**3**

 *0* [s]

**6**

**1.20 1.25 1.30 1.35**

Detection Energy [eV]

**PbS 2.6nm CQDs**

blue arrow is the energy at the half PL intensity from the red side of the PL.

 **4K 60K 100K 290K**

(a) (b)

investigated are ternary core or c/sh QDs, having a general formula PbSexS1-x/PbSeyS1-y, covering the following cases: (a) x = y = 1 or x = y = 0 refers to a simple core PbSe or PbS, respectively; (b) 0 < x = y < 1 is a homogenous alloy core; (c) x = 1 and y = 0 is a simple PbSe/PbS c/sh; (d) x = 1 (y = 1) and y 0 (x 0) is a complex c/sh QD, when either the core or the shell has a homogenous alloyed composition. A schematic drawing of a ternary QD is shown in Figure 13. *Rc* and *Rs* designate the radius of a core and a c/sh QD, respectively.

Fig. 13. Schematic drawing of a spherical c/sh PbSe*x*S1-*x*/PbSe*y*S1-*y* QD. *Rc* and *Rs* are the core and the total radii, respectively. Radial variation of the bulk material parameters in spherical PbSe/PbS QD (*Rc*=2 nm, *Rs*=3 nm).

Three-dimensional plots of the electron and hole distribution functions on (111) cut-plane for a pure core (Panel (a)), c/sh QDs (Panels (b) and (c)) and c/a-sh (Panels (d) and (e)) of equivalent *Rs,* are shown in Figure 14. Panels (f) and (g) show the electron and the hole distribution for a c/a-sh with *Rc*=3 nm and *Rs*=5 nm. In the case of a pure core structure, the distribution of electron and hole is virtually identical, thus Panel (a) describes either one of the carriers. The choice of the (111) plane is made for the calculation convenience only, and is equivalent to choosing any other crystallographic plane for the distribution representation, since the ground state wavefunctions are spherically symmetric. These plots reveal a distinct trend, in which the lowest energy hole state, 1/2, 1 (1/2 denotes the total angular momentum *j* of the state, and ±1 corresponds to the parity *π*), is more delocalized with respect to its counter partner, the lowest energy electron state 1/2, 1 , in c/sh and c/a-sh QDs, characteristic of *quasi*-type II configuration at the band edge. This electronic distribution explains the experimental observations shown in Figure 2 of the gradual redshift of the absorption spectra with the increase of the shell width . The electron and hole spatial distribution functions in c/sh CQDs shown in Figure 14(b) and 14(c) suggest that the distribution of both carriers is similar to that of a c/a-sh structure (Figure 14(d) and 14(e)); however, the distribution differs from the case of a simple core QD of comparable total size (Figure 14(a)). The calculations show that for heterostructured particles with *Rc*< 3 nm the energies of the lowest lying electron levels exceed the potential barrier height (either abrupt or gradual), located at the interface between the core and the shell, thus significantly reducing the effect of quantum confinement induced by the shell layer. On the other hand, in the case of larger c/sh or c/a-sh particles the electron energy is lower than the barrier height, thus enforcing the confinement (and subsequently the localization) of the electron in

Temperature-Dependent Optical Properties of Colloidal IV-VI

the energy levels of the carriers.

nm.

Quantum Dots, Composed of Core/Shell Heterostructures with Alloy Components 81

Figure 14(h) displays the calculated energy of two lowest energy states versus the *Rs* of core and c/sh QDs with *Rs* / *Rc* =3/2. This Figure reveals a pronounced influence of the shell on

In the case of a c/sh structure, both the electron and hole levels are lowered in energy relative to a core structure of the same size, with a larger influence on the hole levels. In the framework of this model the energy levels of c/a-sh structures are almost identical to those of c/sh, hence are not shown here. This finding is consistent with the experimental observation of the red-shift in the emission energy of the c/sh and c/a-sh heterostructures, relative to the cores of corresponding size. The theoretical 1/2, 1 1/2, 1 transition energies (which is the first excitonic transition) are listed in Table 1 and are compared with the experimental absorption band edge energies, with a close agreement for QDs with *Rs*> 1.5 nm (Apparently, the accuracy of the model is not satisfactory for very small sizes due to the breakdown of the major assumption that the envelope function is slowly varying on the scale of the unit cell). The model reproduced the band edge energies of the QDs with relatively close agreement with the experiment, as well as predicted varying delocalization extent of the electrons in the lowest conduction band. The explanation of the reported variation of various physical properties of c/sh and c/a-sh heterostructures would demand further theoretical considerations (e.g., mass anisotropy, exchange interactions), which are

beyond the scope of the discussed model, however, will be done in the future.

and composition of QD with *Rs* are shown in Figure 15.

Energy (eV)

0.3

0.4

0.5

Evolution of the ten lowest conduction and valence band levels as a function of structure

Fig. 15. Evolution of the energy of the conduction and valence band energy levels through a series of composition and structural changes, while maintaining a constant QD radius of 2.5

*x y x*

S1-*y* PbSe*x*

S1-*x*

PbSe PbS

0 0.5 1 1 0.5 0 1 0.5 0

Composition

PbSe*x* S1-*x* PbSe/PbSe /PbS *y*

PbS PbSe/PbS

First, the QD structure evolves from PbS core to PbSe core via the intermediate alloyed PbSe*x*S1-*x* structures (Left Panel). Next, the QD is divided into a 3 nm PbSe core and a 2 nm

the core region ( *cf.* Figure 14(b) and 14(f) ). However, the energy difference between the valence band edges of PbSe and PbS is almost an order of magnitude smaller than that between the conduction band edges, hence the hole distribution is influenced by the shell to a much lesser extent (*cf*. Figure 14(c) and 14(g) ).

Fig. 14. Probability density on the (111) cut-plane for electron and hole in (a) PbSe QDs, *Rs=*2.4 nm. (b) Electron and (c) hole in PbSe/PbS CQDs, *Rc*=1.5 nm, *Rs*=2.4 nm. (d) Electron and (e) hole in c/a-sh QDs, *Rc* =1.6 nm, *Rs* =2.4 nm. (f) Electron and (g) hole in c/a-shQDs, *Rc* =3 nm, *Rs* =5 nm (Both dashed circles, outer and inner, represent the external particle boundary and the core/shell interface location, respectively). The insets marked VB (valence band) and CB (conduction band) schematically represent the radial potential energy profile used in the calculation in each case. (h) Energy as a function of *Rs* of two lowest states of electron and hole in PbSe core and PbSe/PbS c/sh QDs with *Rs* / *Rc* =3/2.

the core region ( *cf.* Figure 14(b) and 14(f) ). However, the energy difference between the valence band edges of PbSe and PbS is almost an order of magnitude smaller than that between the conduction band edges, hence the hole distribution is influenced by the shell to

Fig. 14. Probability density on the (111) cut-plane for electron and hole in (a) PbSe QDs, *Rs=*2.4 nm. (b) Electron and (c) hole in PbSe/PbS CQDs, *Rc*=1.5 nm, *Rs*=2.4 nm. (d) Electron and (e) hole in c/a-sh QDs, *Rc* =1.6 nm, *Rs* =2.4 nm. (f) Electron and (g) hole in c/a-shQDs, *Rc* =3 nm, *Rs* =5 nm (Both dashed circles, outer and inner, represent the external particle boundary and the core/shell interface location, respectively). The insets marked VB (valence band) and CB (conduction band) schematically represent the radial potential energy profile used in the calculation in each case. (h) Energy as a function of *Rs* of two lowest states of

electron and hole in PbSe core and PbSe/PbS c/sh QDs with *Rs* / *Rc* =3/2.

a much lesser extent (*cf*. Figure 14(c) and 14(g) ).

Figure 14(h) displays the calculated energy of two lowest energy states versus the *Rs* of core and c/sh QDs with *Rs* / *Rc* =3/2. This Figure reveals a pronounced influence of the shell on the energy levels of the carriers.

In the case of a c/sh structure, both the electron and hole levels are lowered in energy relative to a core structure of the same size, with a larger influence on the hole levels. In the framework of this model the energy levels of c/a-sh structures are almost identical to those of c/sh, hence are not shown here. This finding is consistent with the experimental observation of the red-shift in the emission energy of the c/sh and c/a-sh heterostructures, relative to the cores of corresponding size. The theoretical 1/2, 1 1/2, 1 transition energies (which is the first excitonic transition) are listed in Table 1 and are compared with the experimental absorption band edge energies, with a close agreement for QDs with *Rs*> 1.5 nm (Apparently, the accuracy of the model is not satisfactory for very small sizes due to the breakdown of the major assumption that the envelope function is slowly varying on the scale of the unit cell). The model reproduced the band edge energies of the QDs with relatively close agreement with the experiment, as well as predicted varying delocalization extent of the electrons in the lowest conduction band. The explanation of the reported variation of various physical properties of c/sh and c/a-sh heterostructures would demand further theoretical considerations (e.g., mass anisotropy, exchange interactions), which are beyond the scope of the discussed model, however, will be done in the future.

Evolution of the ten lowest conduction and valence band levels as a function of structure and composition of QD with *Rs* are shown in Figure 15.

Fig. 15. Evolution of the energy of the conduction and valence band energy levels through a series of composition and structural changes, while maintaining a constant QD radius of 2.5 nm.

First, the QD structure evolves from PbS core to PbSe core via the intermediate alloyed PbSe*x*S1-*x* structures (Left Panel). Next, the QD is divided into a 3 nm PbSe core and a 2 nm

Temperature-Dependent Optical Properties of Colloidal IV-VI

Quantum Dots, Composed of Core/Shell Heterostructures with Alloy Components 83

Fig. 17. Calculated (empty circles) and the corresponding experimental (filled circles) values

The synthesis of core PbSe CQDs followed a modified procedure to that given by Murray et al.50 , following the procedure given in19 and including the preceding stages: (1) 0.71 gr of lead(II) acetate trihydrate [Pb-ac] (Pb[CH3COO]2 · 3H2O, GR, Merck) were dissolved in a solution composed of 2 mL diphenyl ether [PhEt] (C6H5OC6H5 , 99%, Aldrich), 1.5 mL oleic acid (OA) (CH3(CH2)7 CHCH(CH2)7COOH, 99.8%, Aldrich) and 8 mL TOP ((C8H17)3P, Tech, Aldrich), under standard inert conditions in the glove box, and were inserted into a threeneck flask (flask I); (2) 10 mL of PhEt were inserted into a three-neck flask (flask II) under the inert conditions of a glove box; (3) both flasks were taken out of the glove box, placed on a Schlenk line and heated under a vacuum to 100 - 120°C for an hour; (4) flask I was cooled to 45°C, while flask II was heated to 180 -210°C, both under a fledging of an argon gas; (5) 0.155 gr of selenium powder (Se, 99.995%, Aldrich) was dissolved in 2.0 mL TOP, forming a TOP:Se solution, under standard inert conditions of a glove box. Then, 1.7 mL of this solution was injected into flask I on the Schlenk line; (6) the content of flask I, containing the reaction precursors, was injected rapidly into the PhEt solution in flask II, reducing its temperature to 100 - 130°C, leading to the formation of PbSe CQDs within the first 15 min of the reaction. The described procedure produced nearly mono-dispersed CQDs with < 8% size distribution, with average size between 3 and 9 nm, controlled by the temperature and

The preparation of PbSe/PbS c/sh CQDs by a two-injection process19 begins with formation of core PbSe CQDs and their isolation from the initial reaction solution, according to the procedure described previolsy. Those core CQDs were re-dissolved in chloroform solution, forming a solution of 50 mg/mL weight concentration. The quality of 1.4 mL of TOP was

**5. Synthesis procedures and experimental techniques used for core/shell**

**5.1 Synthesis of PbSe, PbSe/PbS c/sh, PbSe/ PbSexS1-x c/a-sh and PbSey**

of the band gap energies evaluated for several core and c/sh QDs.

**heterostructures with alloy components**

**S1-y/PbSexS1-x a-c/a-sh CQDs**

by the time duration of the reaction.

thick PbSe*y*S1-*y* shell (*i.e., Rc*=3 nm, *Rs*=5 nm). The shell composition then varies from y=1 to y=0, corresponding to a transition from pure PbSe to PbSe/PbS c/sh via intermediate PbSe/PbSe*y*S1-*y* c/sh structures (middle panel). Finally, the composition of the core constituent evolves from x=1 to x=0, corresponding to a transition from PbSe/PbS c/sh to a pure PbS core, which completes the cycle (Right Panel). States of even (+) and odd (-) parity are marked by purple and green lines, respectively. CB and VB correspond to the conduction and valence bands, respectively.

Figure 16 shows the energy levels (blue) and density of states (DOS) (green) of PbS (left panel), PbSe0.5S0.5 (middle panel) and PbSe (right panel) having *Rs*=2.5 nm. The density of states calculated by broadening each energy level by 25 meV Gaussian. The composition is found to have a significant impact on the energy spectrum of spherical QDs (apart from the band gap energy). For instance, when looking at the all three cases of PbSe*x*S1-x core structures of the same size, in PbSe0.5S0.5 the levels are arranged into more dense discrete groups, while in PbS and PbSe QDs they are more evenly distributed, as can be seen in Figure 16.

Fig. 16. Single-particle energy levels (blue) and density of states (green) of PbS (left panel), PbSe0.5S0.5 (middle panel) and PbSe (right panel) QDs having *Rs*=2.5 nm. The density of states calculated by broadening each energy level by 25 meV Gaussian.

It is intuitively clear that the origin of this composition-dependent variance of the DOS (that even reaches degeneracy in some cases) should be linked to some symmetry property of the system. In our case this property is the shape of the energy isosurface (EI) of PbSe*x*S1-*x* bulk materials in the reciprocal space. As mentioned above, in the case of lead chalcogenides the general shape of the EI resembles a spheroid, having its principle axis in the L-direction, and it varies as a function of energy *E***k** .

Calculated values of the fundamental gap energies *Eg* evaluated for various core and c/sh QDs, compared with experimental data are presented in Figure 17 and shows a qualitativly agreement between the theoretical and experimental results.

thick PbSe*y*S1-*y* shell (*i.e., Rc*=3 nm, *Rs*=5 nm). The shell composition then varies from y=1 to y=0, corresponding to a transition from pure PbSe to PbSe/PbS c/sh via intermediate PbSe/PbSe*y*S1-*y* c/sh structures (middle panel). Finally, the composition of the core constituent evolves from x=1 to x=0, corresponding to a transition from PbSe/PbS c/sh to a pure PbS core, which completes the cycle (Right Panel). States of even (+) and odd (-) parity are marked by purple and green lines, respectively. CB and VB correspond to the

Figure 16 shows the energy levels (blue) and density of states (DOS) (green) of PbS (left panel), PbSe0.5S0.5 (middle panel) and PbSe (right panel) having *Rs*=2.5 nm. The density of states calculated by broadening each energy level by 25 meV Gaussian. The composition is found to have a significant impact on the energy spectrum of spherical QDs (apart from the band gap energy). For instance, when looking at the all three cases of PbSe*x*S1-x core structures of the same size, in PbSe0.5S0.5 the levels are arranged into more dense discrete groups, while in PbS and PbSe QDs they are more evenly distributed, as can be seen in

Fig. 16. Single-particle energy levels (blue) and density of states (green) of PbS (left panel), PbSe0.5S0.5 (middle panel) and PbSe (right panel) QDs having *Rs*=2.5 nm. The density of

It is intuitively clear that the origin of this composition-dependent variance of the DOS (that even reaches degeneracy in some cases) should be linked to some symmetry property of the system. In our case this property is the shape of the energy isosurface (EI) of PbSe*x*S1-*x* bulk materials in the reciprocal space. As mentioned above, in the case of lead chalcogenides the general shape of the EI resembles a spheroid, having its principle axis in the L-direction, and

Calculated values of the fundamental gap energies *Eg* evaluated for various core and c/sh QDs, compared with experimental data are presented in Figure 17 and shows a qualitativly

states calculated by broadening each energy level by 25 meV Gaussian.

agreement between the theoretical and experimental results.

it varies as a function of energy *E***k** .

conduction and valence bands, respectively.

Figure 16.

Fig. 17. Calculated (empty circles) and the corresponding experimental (filled circles) values of the band gap energies evaluated for several core and c/sh QDs.

### **5. Synthesis procedures and experimental techniques used for core/shell heterostructures with alloy components**

### **5.1 Synthesis of PbSe, PbSe/PbS c/sh, PbSe/ PbSexS1-x c/a-sh and PbSey S1-y/PbSexS1-x a-c/a-sh CQDs**

The synthesis of core PbSe CQDs followed a modified procedure to that given by Murray et al.50 , following the procedure given in19 and including the preceding stages: (1) 0.71 gr of lead(II) acetate trihydrate [Pb-ac] (Pb[CH3COO]2 · 3H2O, GR, Merck) were dissolved in a solution composed of 2 mL diphenyl ether [PhEt] (C6H5OC6H5 , 99%, Aldrich), 1.5 mL oleic acid (OA) (CH3(CH2)7 CHCH(CH2)7COOH, 99.8%, Aldrich) and 8 mL TOP ((C8H17)3P, Tech, Aldrich), under standard inert conditions in the glove box, and were inserted into a threeneck flask (flask I); (2) 10 mL of PhEt were inserted into a three-neck flask (flask II) under the inert conditions of a glove box; (3) both flasks were taken out of the glove box, placed on a Schlenk line and heated under a vacuum to 100 - 120°C for an hour; (4) flask I was cooled to 45°C, while flask II was heated to 180 -210°C, both under a fledging of an argon gas; (5) 0.155 gr of selenium powder (Se, 99.995%, Aldrich) was dissolved in 2.0 mL TOP, forming a TOP:Se solution, under standard inert conditions of a glove box. Then, 1.7 mL of this solution was injected into flask I on the Schlenk line; (6) the content of flask I, containing the reaction precursors, was injected rapidly into the PhEt solution in flask II, reducing its temperature to 100 - 130°C, leading to the formation of PbSe CQDs within the first 15 min of the reaction. The described procedure produced nearly mono-dispersed CQDs with < 8% size distribution, with average size between 3 and 9 nm, controlled by the temperature and by the time duration of the reaction.

The preparation of PbSe/PbS c/sh CQDs by a two-injection process19 begins with formation of core PbSe CQDs and their isolation from the initial reaction solution, according to the procedure described previolsy. Those core CQDs were re-dissolved in chloroform solution, forming a solution of 50 mg/mL weight concentration. The quality of 1.4 mL of TOP was

Temperature-Dependent Optical Properties of Colloidal IV-VI

<N0>= *jp* **.**

absorption was more than 20%.

**5.4 Storage conditions**

Quantum Dots, Composed of Core/Shell Heterostructures with Alloy Components 85

The absorption spectra of the samples were recorded on a JASCO V-570 UV-VIS-NIR spectrometer. The cw-PL spectra were obtained by exciting the samples with a tunable Ti:Sapphire laser, (*Eexc* = 1.48-1.80 eV). The PL spectra of the materials studied were recorded at a temperature range of 1.4 K to 300 K, while immersing the samples in a variable temperature Janis cryostat, and detecting the emission with an Acton Spectrapro 2300i monochromator, which was equipped with a cooled InGaAs CCD or cooled Ge photo detector. The transient PL curves were recorded by exciting the samples with a Nd:YAG laser, (*Eexc* = 1.17 eV). The measurements utilized a laser flux <0.1 mJ/cm2, corresponding to a photon fluence of jp~1011 photons/cm2 per pulse. Considering the absorption cross-section of 0~10-15 cm2, measured in reference,14 the number of photo-generated excitons is given by

transient-PL curves were monitored by a photon multiplier tube, Hamamatsu NIR-PMT H10330-75. The measurements were carried out at temperature range from 1.4 K to 300 K. The PL quantum yield was measured utilizing integrating sphere technique described by Friend.51 A solution of CQDs was placed inside an integrating sphere and excited by a monochromatic light. Luminescence spectra were detected by a fiber-coupled spectrometer equipped with a liquid nitrogen-cooled Ge photo detector. The entire system response was normalized against a calibrated detector, and care was taken to ensure that the sample

The PbSeyS1-y/PbSexS1-x a-c/a-sh (0<x<1;0<y<1) CQDs were stored either in a various solutions (chloroform, hexane) or were embedded in a polymer film or dissolved in a GS (2,2,4,4,6,8,8,-heptamethyl-nonane) for the optical measurements. The polymer-CQDs solution was prepared by mixing PbSe CQDs in chloroform solution with poly-methylmethacrylate (PMMA) ([–CH 2 C(CH 3 )(CO 2 CH 3 )–] *n* , analytical grade, Aldrich) polymer solution. The resultant mixture was spread on a quartz substrate and dried over 24 h to a uniform film. The stability of these CQDs was examined over a period of time by recording the absorption spectra and following the consistency of the low exciton energy. Plots of this exciton energy versus time suggest that the exciton energy in the core samples is blueshifted by ~ 400 meV over a period of days for the CQDs kept in a chloroform solution. Such a blue-shift, however, occurs over months for the samples kept as dry powders. On the other hand, the energy shift is smaller for the PbSe/PbS c/sh samples, and even nearly disappeared for the CQDs coated with three PbS shell monolayers. It is presumed that the exciton energy blue-shift is due to oxidation of the surface, and a decrease of the effective size of the core. Obviously, the penetration of oxygen through the PbS shell is reduced with the growth of the shell width. Furthermore, storage of the CQDs in a nitrogen environment nearly eliminates any spectral drift over a period of months, even extending to two or three years. It should be noted that most of the optical measurements were carried out at cryogenic temperatures, inducing a He gas environment around the samples. But unpublished results determined degradation of the samples when exposed to intense pulsed

Unique alloyed c/sh heterostructures, such as PbSeyS1-y /PbSexS1-x, (0<x≤1; 0<y≤1) were developed, offering good crystallographic and dielectric match at the c/sh interface,

UV radiation, which is avoided completely in the current study.

**6. Conclusion and future directions**

*0*, and estimated to be 10-4<<1, ensuring the generation of single excitons. The

then added to the CQDs solution, while the chloroform molecules were removed by distillation under vacuum and heating at 60°C. In parallel, 0.2 gr of a Pb precursor, Pb-ac, was dissolved in a mixture of 2 mL PhEt, 1.5 mL of OA, and 8 mL of TOP, heated to 120°C for an hour, and then cooled to 45°C. Also, 0.03–0.10 gr of sulphur (S, 99.99 + %, Aldrich) was dissolved in 0.3 mL of TOP and was premixed with a PbSe core CQDs in a TOP solution. This mixture was injected into the Pb-ac solution. All reagents were then injected into a PhEt mother solution and kept on a Schlenk line at 180°C, causing a reduction in temperature of the mother solution to 120°C. The indicated chemical portions caused the precipitation of 1–3 monolayers of PbS shell over the PbSe core surface within the first 15 min of the reaction.

The preparation of PbSe/PbSe*x*S1-x c/a–sh and PbSexS1-x/PbSeyS1-y a-c/a-sh structures19 is nearly identical to that of the core PbSe CQDs, described previosly using a single injection of the precursors into a single round flask. However, step (5) was altered by the use of an alternative chalcogen precursor stock solution. A stock solution of Se and S was prepared by mixing 0.15 gr Se dissolved in 1.4 mL TOP, with 0.03–0.10 gr S dissolved in 0.3 mL TOP. The amount of S in the new stock solution corresponded to a stoichiometric amount of 1–2 ML of the PbS compound. Thus, the mole ratio of the precursors Pb:Se:S ranged from 1:1:0.5 to 1:1:1.3. Aliquots were drawn periodically from the mother solutions while quenching process to RT terminated the CQDs' growth. They were isolated from the aliquots solution by the addition of methanol, and by centrifugation. The isolated CQDs were further purified by dissolving them in chloroform, followed by filtering several times through a 0.02 micron membrane. A preliminary injection of Pb/Se/S ions ratio of 1/1/0.5 led to the nucleation of a pure PbSe core, due to the faster reactivity of the Se precursors at the nucleation stage. However, the increase of the S/Se ratio (S/Se> 1.5/1) enabled an immediate integration of both elements with the nuclei (monitored already in the first aliquot). Further aliquots revealed a gradient increase of the S/Se ratio when moving from the interface toward the exterior surface. For simplicity, the samples were labeled as PbSeyS1-y/PbSexS1-x.

### **5.2 Synthesis of PbS CQDs**

PbS CQDs were prepared according to literature procedure.43 The lead oleate precursor was prepared by heating 0.09 g PbO in 4 ml OA under N2 at 1200C for 1h. A solution of 42 l bis(trimethylsilyl)sulfide (TMS ) in 2 ml octadecene (ODE) was injected into the vigorously stirring lead oleate solution. The final particle size was controlled by injection temperature (120-150 0C), where lower temperature leads to smaller sizes, and by varying the Pb:OA molar ratios (2:32 to 2:4, where the total volume of lead oleate solution was kept at 4 ml by dilution with ODE). The reaction was quenched by cooling it to RT and the CQDs were precipitated with acetone, then subsequently redispersed in chloroform and precipitated again.

### **5.3 Experimenthal methods**

The morphology and crystallography of the colloidal CQDs were examined by X-ray diffraction, TEM, HR-TEM and SAED. The TEM specimens were prepared by injecting small liquid droplets of the solution on a copper grid (300 mesh) coated with amorphous carbon film and then dried at room temperature. The elemental analysis were examined by EDAX, and inductively coupled plasma atomic emission spectroscopy (ICP-AES).

The absorption spectra of the samples were recorded on a JASCO V-570 UV-VIS-NIR spectrometer. The cw-PL spectra were obtained by exciting the samples with a tunable Ti:Sapphire laser, (*Eexc* = 1.48-1.80 eV). The PL spectra of the materials studied were recorded at a temperature range of 1.4 K to 300 K, while immersing the samples in a variable temperature Janis cryostat, and detecting the emission with an Acton Spectrapro 2300i monochromator, which was equipped with a cooled InGaAs CCD or cooled Ge photo detector. The transient PL curves were recorded by exciting the samples with a Nd:YAG laser, (*Eexc* = 1.17 eV). The measurements utilized a laser flux <0.1 mJ/cm2, corresponding to a photon fluence of jp~1011 photons/cm2 per pulse. Considering the absorption cross-section of 0~10-15 cm2, measured in reference,14 the number of photo-generated excitons is given by <N0>= *jp* **.** *0*, and estimated to be 10-4<<1, ensuring the generation of single excitons. The transient-PL curves were monitored by a photon multiplier tube, Hamamatsu NIR-PMT H10330-75. The measurements were carried out at temperature range from 1.4 K to 300 K. The PL quantum yield was measured utilizing integrating sphere technique described by Friend.51 A solution of CQDs was placed inside an integrating sphere and excited by a monochromatic light. Luminescence spectra were detected by a fiber-coupled spectrometer equipped with a liquid nitrogen-cooled Ge photo detector. The entire system response was normalized against a calibrated detector, and care was taken to ensure that the sample absorption was more than 20%.

### **5.4 Storage conditions**

84 Fingerprints in the Optical and Transport Properties of Quantum Dots

then added to the CQDs solution, while the chloroform molecules were removed by distillation under vacuum and heating at 60°C. In parallel, 0.2 gr of a Pb precursor, Pb-ac, was dissolved in a mixture of 2 mL PhEt, 1.5 mL of OA, and 8 mL of TOP, heated to 120°C for an hour, and then cooled to 45°C. Also, 0.03–0.10 gr of sulphur (S, 99.99 + %, Aldrich) was dissolved in 0.3 mL of TOP and was premixed with a PbSe core CQDs in a TOP solution. This mixture was injected into the Pb-ac solution. All reagents were then injected into a PhEt mother solution and kept on a Schlenk line at 180°C, causing a reduction in temperature of the mother solution to 120°C. The indicated chemical portions caused the precipitation of 1–3 monolayers of PbS shell over the PbSe core surface within the first 15

The preparation of PbSe/PbSe*x*S1-x c/a–sh and PbSexS1-x/PbSeyS1-y a-c/a-sh structures19 is nearly identical to that of the core PbSe CQDs, described previosly using a single injection of the precursors into a single round flask. However, step (5) was altered by the use of an alternative chalcogen precursor stock solution. A stock solution of Se and S was prepared by mixing 0.15 gr Se dissolved in 1.4 mL TOP, with 0.03–0.10 gr S dissolved in 0.3 mL TOP. The amount of S in the new stock solution corresponded to a stoichiometric amount of 1–2 ML of the PbS compound. Thus, the mole ratio of the precursors Pb:Se:S ranged from 1:1:0.5 to 1:1:1.3. Aliquots were drawn periodically from the mother solutions while quenching process to RT terminated the CQDs' growth. They were isolated from the aliquots solution by the addition of methanol, and by centrifugation. The isolated CQDs were further purified by dissolving them in chloroform, followed by filtering several times through a 0.02 micron membrane. A preliminary injection of Pb/Se/S ions ratio of 1/1/0.5 led to the nucleation of a pure PbSe core, due to the faster reactivity of the Se precursors at the nucleation stage. However, the increase of the S/Se ratio (S/Se> 1.5/1) enabled an immediate integration of both elements with the nuclei (monitored already in the first aliquot). Further aliquots revealed a gradient increase of the S/Se ratio when moving from the interface toward the

exterior surface. For simplicity, the samples were labeled as PbSeyS1-y/PbSexS1-x.

with acetone, then subsequently redispersed in chloroform and precipitated again.

and inductively coupled plasma atomic emission spectroscopy (ICP-AES).

PbS CQDs were prepared according to literature procedure.43 The lead oleate precursor was prepared by heating 0.09 g PbO in 4 ml OA under N2 at 1200C for 1h. A solution of 42 l bis(trimethylsilyl)sulfide (TMS ) in 2 ml octadecene (ODE) was injected into the vigorously stirring lead oleate solution. The final particle size was controlled by injection temperature (120-150 0C), where lower temperature leads to smaller sizes, and by varying the Pb:OA molar ratios (2:32 to 2:4, where the total volume of lead oleate solution was kept at 4 ml by dilution with ODE). The reaction was quenched by cooling it to RT and the CQDs were precipitated

The morphology and crystallography of the colloidal CQDs were examined by X-ray diffraction, TEM, HR-TEM and SAED. The TEM specimens were prepared by injecting small liquid droplets of the solution on a copper grid (300 mesh) coated with amorphous carbon film and then dried at room temperature. The elemental analysis were examined by EDAX,

min of the reaction.

**5.2 Synthesis of PbS CQDs**

**5.3 Experimenthal methods**

The PbSeyS1-y/PbSexS1-x a-c/a-sh (0<x<1;0<y<1) CQDs were stored either in a various solutions (chloroform, hexane) or were embedded in a polymer film or dissolved in a GS (2,2,4,4,6,8,8,-heptamethyl-nonane) for the optical measurements. The polymer-CQDs solution was prepared by mixing PbSe CQDs in chloroform solution with poly-methylmethacrylate (PMMA) ([–CH 2 C(CH 3 )(CO 2 CH 3 )–] *n* , analytical grade, Aldrich) polymer solution. The resultant mixture was spread on a quartz substrate and dried over 24 h to a uniform film. The stability of these CQDs was examined over a period of time by recording the absorption spectra and following the consistency of the low exciton energy. Plots of this exciton energy versus time suggest that the exciton energy in the core samples is blueshifted by ~ 400 meV over a period of days for the CQDs kept in a chloroform solution. Such a blue-shift, however, occurs over months for the samples kept as dry powders. On the other hand, the energy shift is smaller for the PbSe/PbS c/sh samples, and even nearly disappeared for the CQDs coated with three PbS shell monolayers. It is presumed that the exciton energy blue-shift is due to oxidation of the surface, and a decrease of the effective size of the core. Obviously, the penetration of oxygen through the PbS shell is reduced with the growth of the shell width. Furthermore, storage of the CQDs in a nitrogen environment nearly eliminates any spectral drift over a period of months, even extending to two or three years. It should be noted that most of the optical measurements were carried out at cryogenic temperatures, inducing a He gas environment around the samples. But unpublished results determined degradation of the samples when exposed to intense pulsed UV radiation, which is avoided completely in the current study.

### **6. Conclusion and future directions**

Unique alloyed c/sh heterostructures, such as PbSeyS1-y /PbSexS1-x, (0<x≤1; 0<y≤1) were developed, offering good crystallographic and dielectric match at the c/sh interface,

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Quantum Dots, Composed of Core/Shell Heterostructures with Alloy Components 87

[2] Peng, X.; Manna, L.; Yang, W.; Wickham, J.; Scher, E.; Kadavanich, A.; Alivisatos, A. P.,

[3] Akamatsu, K.; Tsuruoka, T.; Nawafune, H., Band Gap Engineering of CdTe Nanocrystals

[4] Ma, W.; Luther, J. M.; Zheng, H.; Wu, Y.; Alivisatos, A. P., Photovoltaic Devices Employing Ternary PbSxSe1-x Nanocrystals. *Nano Letters* 2009, *9* (4), 1699-1703. [5] Bailey, R. E.; Nie, S., Alloyed Semiconductor Quantum Dots: Tuning the Optical

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[11] Oron, D.; Kazes, M.; Banin, U., Multiexcitons in type-II colloidal semiconductor

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regulating carriers' delocalization and/or charge separation by tunability of the band off-set, showing an exceptionally high emission quantum yield, chemical stability, and an option to stabilize the emission intensity (blinking free behavior), as well as sustain the biexciton lifetime over a nanosecond. The last can be of a valuable benefit in the use of CQDs in gain devices and photovoltaic cells.

 A smooth potential at the c/sh interface was applied here for the determination of the electronic structure of IV-VI CQDs, using a **k p** model, covered a wide physical aspects, including an effective mass anisotropy, dielectric variation between the constituents, showing a ground for tailoring heterostructures with the desired composition and optical properties.

A thorough investigation of the optical properties was performed by following variable temperature cw- and transient spectrally resolved PL spectra, exploring energy shift, band edge temperature coefficient, alleviation of a dark-bright splitting (or exchange interaction), valley-valley interaction, emission quantum yield, and radiative lifetime of the PbSeyS1-y /PbSexS1-x heterostructures, in comparison with the existing properties of the primary PbSe core CQDs. Temporally and spectrally resolved PL spectra provide more-systematic evidence of the two emissive centers nature. The results reflect the uniqueness of the electronic properties of the heterostructures, controlled by shell width and alloyed composition.

The discussed heterostructures could be of significant importance in applications where the CQDs' size is restricted, *e.g.*, biological markers or self-assembled CQDs in opto-electronic devices, while at the same time, there are stringent demands regarding the optical tunability. We showed that the restriction can be overcome by the discussed new strategies gaining property control using: (a) alloyed ternary or quaternary compounds, when all elements can be either distributed homogeneously or exhibit a graded composition along a selective direction; (b) c/sh heterostructures, comprised of a semiconductor core, covered by a shell, of another semiconductor, when the band-edge off-set at the core/shell interface, can be tuned from a type-I (when shell band-edge is rapping that of the core), through *quasi*type-II, to a type-II (when, band-edge of the constituents are staggered) alignment. Moreover, one of the constituents (core or shell) may have alloyed composition.

### **7. Acknowledgment**

The authors thank G. Kventsel for helpful discussions, providing many insightful comments, A. Bartnik and F. Wise for helpful discussions and guidance on the theoretical model, A. Efros for the useful scientific discussions, and O. Solomeshch for assistance in the quantum yield measurements. The authors acknowledge support from the Israel Science Foundation (Projects No. 1009/07 and No. 1425/04), the USA-Israel Binational Science Foundation (No. 2006-225), and the Ministry of Science (No. 3-896).

### **8. References**

[1] Steigerwald, M. L.; Brus, L. E., Semiconductor crystallites: a class of large molecules. *Accounts of Chemical Research* 1990, *23* (6), 183-188.

regulating carriers' delocalization and/or charge separation by tunability of the band off-set, showing an exceptionally high emission quantum yield, chemical stability, and an option to stabilize the emission intensity (blinking free behavior), as well as sustain the biexciton lifetime over a nanosecond. The last can be of a valuable benefit in the use of CQDs in gain

 A smooth potential at the c/sh interface was applied here for the determination of the electronic structure of IV-VI CQDs, using a **k p** model, covered a wide physical aspects, including an effective mass anisotropy, dielectric variation between the constituents, showing a ground for tailoring heterostructures with the desired composition and optical

A thorough investigation of the optical properties was performed by following variable temperature cw- and transient spectrally resolved PL spectra, exploring energy shift, band edge temperature coefficient, alleviation of a dark-bright splitting (or exchange interaction), valley-valley interaction, emission quantum yield, and radiative lifetime of the PbSeyS1-y /PbSexS1-x heterostructures, in comparison with the existing properties of the primary PbSe core CQDs. Temporally and spectrally resolved PL spectra provide more-systematic evidence of the two emissive centers nature. The results reflect the uniqueness of the electronic properties of the heterostructures, controlled by shell width and alloyed

The discussed heterostructures could be of significant importance in applications where the CQDs' size is restricted, *e.g.*, biological markers or self-assembled CQDs in opto-electronic devices, while at the same time, there are stringent demands regarding the optical tunability. We showed that the restriction can be overcome by the discussed new strategies gaining property control using: (a) alloyed ternary or quaternary compounds, when all elements can be either distributed homogeneously or exhibit a graded composition along a selective direction; (b) c/sh heterostructures, comprised of a semiconductor core, covered by a shell, of another semiconductor, when the band-edge off-set at the core/shell interface, can be tuned from a type-I (when shell band-edge is rapping that of the core), through *quasi*type-II, to a type-II (when, band-edge of the constituents are staggered) alignment.

The authors thank G. Kventsel for helpful discussions, providing many insightful comments, A. Bartnik and F. Wise for helpful discussions and guidance on the theoretical model, A. Efros for the useful scientific discussions, and O. Solomeshch for assistance in the quantum yield measurements. The authors acknowledge support from the Israel Science Foundation (Projects No. 1009/07 and No. 1425/04), the USA-Israel Binational Science

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**4**

*Italy*

**Optical Properties of**

Giovanni Morello

**Spherical Colloidal Nanocrystals**

*1Nanoscience Institute of CNR, National Nanotechnology Laboratory (NNL), 2Center for Biomolecular Nanotechnologies @UNILE, IIT, Arnesano (LE),*

The desire to fabricate materials with novel or improved properties is a powerful stimulus for the development of materials science. Thermal and electrical conduction, optical response, energy conversion and storage are just a few of the large number of properties underwent a very fast evolution in the last decade thank to the birth of a new branch of materials science and technology defined "Nanotechnology". Nanotechnology includes the totality of the physical, chemical, biological and engineering knowledge involving artificial structures whose properties are controlled at the nanometer level. Among the multitude of nanomaterials created, a particular class of them is becoming very popular and represents nowadays the most fascinating and potentially revolutionary inorganic semiconductor structure, which is the family of the colloidal quantum dots. They are often referred to as "nanocrystals" and the colloidal definition reveal their chemical origin. Actually, the chemical synthesis currently represents the most effective way to obtain high quality (in terms of size control, narrow size distribution, good crystalline structure and high optical performances) nano-objects on a gram scale which can be handled as ordinary chemical substances and implemented in several opto-electronic devices as well as biological ambient.  Chemical syntheses allow for the fabrication of nanocrystals (NCs) with nearly atomic precision. They are currently prepared in a variety of compositions as nearly spherical particles (Peng et al., 2000), elongated nanorods (Krahne et al., 2011), and other more complex structures like tetrapods (Fiore et al., 2009) and octapods (Miszta et al., 2011; Zhang et al., 2011). Moreover, the sophistication with which inorganic nanoparticles can be prepared has inspired the creation of multi-combined nano-systems having different compositions ranging from the all-semiconductor to hybrid semiconductor-metal nanoparticles possessing on demand properties in terms of electronic levels energy. As a consequence, the in-depth knowledge of the electronic structure of nanocrystals became of fundamental importance and a huge effort on the theoretical and experimental point of view

has been made in the two last decades dedicated to their study and comprehension.

**1. Introduction**

## **Optical Properties of Spherical Colloidal Nanocrystals**

Giovanni Morello

*1Nanoscience Institute of CNR, National Nanotechnology Laboratory (NNL), 2Center for Biomolecular Nanotechnologies @UNILE, IIT, Arnesano (LE), Italy*

### **1. Introduction**

90 Fingerprints in the Optical and Transport Properties of Quantum Dots

[50] Murray, C. B.; Shouheng, S.; Gaschler, W.; Doyle, H.; Betley, T. A.; Kagan, C. R.,

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2001, *45*, 47.

*Materials* 1997, *9* (3), 230-232.

Colloidal synthesis of nanocrystals and nanocrystal superlattices. *IBM J. Res. & Dev.*

determination of external photoluminescence quantum efficiency. *Advanced*

The desire to fabricate materials with novel or improved properties is a powerful stimulus for the development of materials science. Thermal and electrical conduction, optical response, energy conversion and storage are just a few of the large number of properties underwent a very fast evolution in the last decade thank to the birth of a new branch of materials science and technology defined "Nanotechnology". Nanotechnology includes the totality of the physical, chemical, biological and engineering knowledge involving artificial structures whose properties are controlled at the nanometer level. Among the multitude of nanomaterials created, a particular class of them is becoming very popular and represents nowadays the most fascinating and potentially revolutionary inorganic semiconductor structure, which is the family of the colloidal quantum dots. They are often referred to as "nanocrystals" and the colloidal definition reveal their chemical origin. Actually, the chemical synthesis currently represents the most effective way to obtain high quality (in terms of size control, narrow size distribution, good crystalline structure and high optical performances) nano-objects on a gram scale which can be handled as ordinary chemical substances and implemented in several opto-electronic devices as well as biological ambient.  Chemical syntheses allow for the fabrication of nanocrystals (NCs) with nearly atomic precision. They are currently prepared in a variety of compositions as nearly spherical particles (Peng et al., 2000), elongated nanorods (Krahne et al., 2011), and other more complex structures like tetrapods (Fiore et al., 2009) and octapods (Miszta et al., 2011; Zhang et al., 2011). Moreover, the sophistication with which inorganic nanoparticles can be prepared has inspired the creation of multi-combined nano-systems having different compositions ranging from the all-semiconductor to hybrid semiconductor-metal nanoparticles possessing on demand properties in terms of electronic levels energy. As a consequence, the in-depth knowledge of the electronic structure of nanocrystals became of fundamental importance and a huge effort on the theoretical and experimental point of view has been made in the two last decades dedicated to their study and comprehension.

Optical Properties of Spherical Colloidal Nanocrystals 93

depend on the type of restriction in one, two or three dimensions, as well as on the size of the nanostructure. In particular, the quantum phenomena start to be noticeable when the lateral extension of the potential well becomes comparable to the particle wavelength. In order to better understand this concept, let us consider the case of the electrons. As elementary particles they exhibit the wave-particle duality of the matter following the "de Broglie" relation (de Broglie, 1924,1925). When immersed in a solid the electron is treated as a particle having an effective mass m\* accounting for the periodicity of the crystal potential. Its linear momentum p can be written in terms of its wave-like nature, p k , where is the Dirac's constant (the Planck's constant divided by 2π ) and k represents its wavenumber, associated to the de Broglie wavelength λ=2π/k . Electrons in a bulk solid are treated as particles not feeling the borders by imposing the periodic boundary conditions, so that the wavefunctions and the energies are not affected by the real spatial extension of the solid. When the solid dimensions approach the electron wavelength, the permitted wavefunctions and energies undergo a series of restrictions in terms of continuity and absolute values. In few words, the system starts to be considered as "quantized". The simplest example of quantum confinement is an electron enclosed in a one-dimensional quantum box having lateral size a and infinitely high walls. In figure 1 we can see the

situation in terms of energy and wavefunctions.

Fig. 1. Electron in a potential well. Energy levels and wavefunctions.

Schrödinger equation in one dimension (Yoffe, 2001):

equation are in the form:

The problem constitutes a classical textbook case that can be approached by solving the

2 2 *d me* 2 *V dx b*

with V = 0, 0<x<a and V = ∞, x≤0, x≥a. The solutions of this second-order linear differential

( ) . *ikx ikx*

(1)

*x Ae Be* (2)

2

This chapter wants to provide an overview of the main studies carried out on spherical nanocrystals on what concerns their electronic structure and optical properties. Particular attention will be devoted to the review of the role played by defect states (especially surface states) on the final optical performances by means of steady-state and time-resolved spectroscopy. For the sake of clarity, the reported discussions will concern two main kinds of nanocrystals, namely CdTe and CdSe. They represent the ideal cases study, since these two materials have historically been the most studied in the field of semiconductor nanostructures. The reason lies on the fact that they cover great part of the optical properties possessed by a number of different other nanocrystals and moreover they present the two main crystallographic symmetries, namely cubic (CdTe) and wurtzite (CdSe).

The chapter is organized as follows: Section 2 will be dedicated to the basic concepts of the low-dimensional systems. The idea is to provide a comprehensive overview of the physics at the basis of the systems studied. We will start from the most common problem consisting in solving the 1-D Schrödinger equation for an electron in a box, extending the discussion to the types of confinement (1-D, 2-D, 3-D) a nano-object undergoes. The size dependence will be reviewed, by distinguishing the different degrees of confinement. A final part will be devoted to the study of the electronic structure of spherical nanocrystals. Section 3 will provide an extensive discussion about the most important recombination processes occurring in spherical nanocrystals. These processes will be basically separated in two big families: radiative and non radiative processes. In the first case we will deal with relaxation processes involving the emission of one or more photons due to electron-hole annihilation. In the second case the excess energy is released as heat inside the materials and/or by excitation of new electron-hole pairs by Auger-like processes. Since great part of the potential applications of colloidal nanocrystals concerns with light emitting devices, particular emphasis is dedicated to the non radiative pathways limiting their optical performances. A review of the main studies reported in literature will be presented, with particular attention to the parameters affecting them, which are the size, the temperature, the excitation density and the surface quality. About the latter, the impact of surface states on the optical properties of CdSe NCs will be treated by means of a four-level model. A last section is dedicated to the Conclusions.

### **2. Fundamentals of nano-physics**

This first section wants to stress the point that any development in nanoscience necessarily requires an understanding of the physical laws governing the matter at the nanoscale and of how the interplay of the various physical properties of a nanoscopic system translates into some novel behaviour or into a new physical property. In this sense, the section will be an overview of the basic physical laws that govern the nanomaterials, with particular emphasis on quantum dots, being the subject of the chapter.

### **2.1 Quantum confinement on low dimensional systems**

The phenomenon of the quantum confinement (i. e. the size quantization) can be observed in systems where the motion of electrons or other particles (holes, excitons, etc.) is restricted at least in one dimension by some potential energy profile. Such a system is usually referred to as a "low dimensional" system. The energy spectra and the wavefunctions localization

This chapter wants to provide an overview of the main studies carried out on spherical nanocrystals on what concerns their electronic structure and optical properties. Particular attention will be devoted to the review of the role played by defect states (especially surface states) on the final optical performances by means of steady-state and time-resolved spectroscopy. For the sake of clarity, the reported discussions will concern two main kinds of nanocrystals, namely CdTe and CdSe. They represent the ideal cases study, since these two materials have historically been the most studied in the field of semiconductor nanostructures. The reason lies on the fact that they cover great part of the optical properties possessed by a number of different other nanocrystals and moreover they present the two

The chapter is organized as follows: Section 2 will be dedicated to the basic concepts of the low-dimensional systems. The idea is to provide a comprehensive overview of the physics at the basis of the systems studied. We will start from the most common problem consisting in solving the 1-D Schrödinger equation for an electron in a box, extending the discussion to the types of confinement (1-D, 2-D, 3-D) a nano-object undergoes. The size dependence will be reviewed, by distinguishing the different degrees of confinement. A final part will be devoted to the study of the electronic structure of spherical nanocrystals. Section 3 will provide an extensive discussion about the most important recombination processes occurring in spherical nanocrystals. These processes will be basically separated in two big families: radiative and non radiative processes. In the first case we will deal with relaxation processes involving the emission of one or more photons due to electron-hole annihilation. In the second case the excess energy is released as heat inside the materials and/or by excitation of new electron-hole pairs by Auger-like processes. Since great part of the potential applications of colloidal nanocrystals concerns with light emitting devices, particular emphasis is dedicated to the non radiative pathways limiting their optical performances. A review of the main studies reported in literature will be presented, with particular attention to the parameters affecting them, which are the size, the temperature, the excitation density and the surface quality. About the latter, the impact of surface states on the optical properties of CdSe NCs will be treated by means of a four-level model. A last

This first section wants to stress the point that any development in nanoscience necessarily requires an understanding of the physical laws governing the matter at the nanoscale and of how the interplay of the various physical properties of a nanoscopic system translates into some novel behaviour or into a new physical property. In this sense, the section will be an overview of the basic physical laws that govern the nanomaterials, with particular emphasis

The phenomenon of the quantum confinement (i. e. the size quantization) can be observed in systems where the motion of electrons or other particles (holes, excitons, etc.) is restricted at least in one dimension by some potential energy profile. Such a system is usually referred to as a "low dimensional" system. The energy spectra and the wavefunctions localization

main crystallographic symmetries, namely cubic (CdTe) and wurtzite (CdSe).

section is dedicated to the Conclusions.

**2. Fundamentals of nano-physics**

on quantum dots, being the subject of the chapter.

**2.1 Quantum confinement on low dimensional systems**

depend on the type of restriction in one, two or three dimensions, as well as on the size of the nanostructure. In particular, the quantum phenomena start to be noticeable when the lateral extension of the potential well becomes comparable to the particle wavelength. In order to better understand this concept, let us consider the case of the electrons. As elementary particles they exhibit the wave-particle duality of the matter following the "de Broglie" relation (de Broglie, 1924,1925). When immersed in a solid the electron is treated as a particle having an effective mass m\* accounting for the periodicity of the crystal potential. Its linear momentum p can be written in terms of its wave-like nature, p k , where is the Dirac's constant (the Planck's constant divided by 2π ) and k represents its wavenumber, associated to the de Broglie wavelength λ=2π/k . Electrons in a bulk solid are treated as particles not feeling the borders by imposing the periodic boundary conditions, so that the wavefunctions and the energies are not affected by the real spatial extension of the solid. When the solid dimensions approach the electron wavelength, the permitted wavefunctions and energies undergo a series of restrictions in terms of continuity and absolute values. In few words, the system starts to be considered as "quantized". The simplest example of quantum confinement is an electron enclosed in a one-dimensional quantum box having lateral size a and infinitely high walls. In figure 1 we can see the situation in terms of energy and wavefunctions.

Fig. 1. Electron in a potential well. Energy levels and wavefunctions.

The problem constitutes a classical textbook case that can be approached by solving the Schrödinger equation in one dimension (Yoffe, 2001):

$$\frac{d^2\phi}{dx^2} + \frac{2me}{b^2} = V\tag{1}$$

with V = 0, 0<x<a and V = ∞, x≤0, x≥a. The solutions of this second-order linear differential equation are in the form:

$$
\varphi(\mathbf{x}) = A \cdot \mathbf{e}^{ik\mathbf{x}} + B \cdot \mathbf{e}^{-ik\mathbf{x}}.\tag{2}
$$

Optical Properties of Spherical Colloidal Nanocrystals 95

The role of the system size on the confinement energy is played by the 1/a2 factor according to which larger the size a, smaller both the point-zero energy and the spacing in the k-space. This leads to smaller absolute energy spacing, whereas the relative spacing is expressed by

> 2 2 2 2 2 2 1 2 2 ( 1) (2 1), 2 2 *EE n n n n n ma ma*

The wavefunctions look like the standing waves on a string. Figure 1 plots the energies of the allowed states and the relative wavefunctions. They have increasing number of nodes with increasing energy according to the fact that more nodes mean shorter wavelength and

In general, all the quantum confined systems can be classified on the basis of the number of dimensions along which the motion of electrons is coerced. The usual terminology refers to as 1-, 2-, 3-D confinement, in which the more evident effect consists in the modulation of the density of states function and a restriction of the allowed energies. In order to better understand what happens to a system when its dimensions start to shrink, one needs to begin from a non-confined structure, such as a bulk material. Here, one can assume that N electrons are not bound to individual atoms such that they can be considered "free" to move in three directions. If we suppose that, as a first approximation, the electron-electron interactions and the crystal potential are negligible (free electron gas model) (Pines, 1963) we can write the kinetic energy of an electron moving in the solid with velocity v=(v ,v ,v ) x y z

> 1 1 2 222 ( ). 2 2 *E mv m v v v xyz*

On the wavefunction point of view the condition of infinite solid is expressed by imposing the so called boundary conditions. They consist in the continuity of the wavefunction

> (,,) ( ,,) (,,) (, ,) (,,) (,, )

The solution of the 3-D Schrödinger equation gives a factored function, written as the

( , , ) ( ) ( ) ( ) exp( )exp( )exp( ) *xyz*

*x y z x y z A ik x ik y ik z*

*xyz x d yz xyz xy d z xyz xyz d*

*x y z*

(11)

(12)

*p mv k* (13)

(15)

. (14)

:

and increases with increasing n.

higher momentum (i.e., energy).

**2.2 Classification of quantum confined systems**

The corresponding wavevector is derived from the relation

φ(x,y,z) at the border of the real finite solid of dimensions dx, dy, dz:

and the corresponding wavelength λ=2π/ k .

product of three independent functions:

For an infinitely high potential barrier, the electron wavefunction must vanish at the borders of the potential well, such as φ(0)=0 and φ(a)=0 . The condition φ(0)=0 produces B=-A, such that:

$$
\varphi(\mathbf{x}) = A(e^{ik\mathbf{x}} - e^{-ik\mathbf{x}}) = \mathbf{2}iA\sin(k\mathbf{x}).\tag{3}
$$

Substituting φ(x) in the Schrödinger equation one can obtain the following expression for the electron energy:

$$E = \frac{\hbar^2 k^2}{2m}.\tag{4}$$

The condition φ(a) =0 produces the following identity:

$$
\varphi(a) = 2iA\sin(ka) = 0.\tag{5}
$$

The identity is verified if ka = nπ, with n=1, 2, 3, ... The parameter k, therefore, results quantized, and the separation between two consecutive allowed values is Δk=π/a. In terms of energy:

$$E = \frac{\hbar^2 k^2}{2m} = \frac{\hbar^2 \pi^2 n^2}{2ma^2}. \tag{6}$$

The lowest energy for the electron, called Ground State energy, is obtained for n = 1,

$$E = \frac{\hbar^2 \pi^2}{2ma^2}.\tag{7}$$

The expression above defines the minimum energy possessed by a quantized system and is termed "point-zero energy" and constitutes a fascinating manifestation of the Heisenberg Uncertainty Principle. In fact, from

$$
\Delta \mathbf{x} \cdot \Delta p \ge \frac{\hbar}{2} \tag{8}
$$

follows

$$
\Delta p \ge \frac{\hbar}{2\Delta \mathbf{x}} = \frac{\hbar}{a} \tag{9}
$$

since a represents the potential well dimension and Δx ≤ a/2. In terms of averaged kinetic energy:

$$
\left\langle E\_k \right\rangle = \left\langle \frac{p^2}{2m} \right\rangle \ge \frac{(\Delta p)^2}{2m} \Longrightarrow \left\langle E\_k \right\rangle \ge \frac{\hbar^2}{2ma^2}.\tag{10}
$$

This expression gives the theoretical lower limit of the possible kinetic energy value for a quantum particle confined in a box (one-dimensional, in this case) having size a. In the reality we find an actual value about π2 ≈ 10 times larger.

The role of the system size on the confinement energy is played by the 1/a2 factor according to which larger the size a, smaller both the point-zero energy and the spacing in the k-space. This leads to smaller absolute energy spacing, whereas the relative spacing is expressed by

$$E\_{n+1} - E\_n = \frac{\hbar^2 \pi^2}{2ma^2} \left[ \left( n + 1 \right)^2 - n^2 \right] = \frac{\hbar^2 \pi^2}{2ma^2} (2n + 1) \text{.}\tag{11}$$

and increases with increasing n.

94 Fingerprints in the Optical and Transport Properties of Quantum Dots

For an infinitely high potential barrier, the electron wavefunction must vanish at the borders of the potential well, such as φ(0)=0 and φ(a)=0 . The condition φ(0)=0 produces B=-A, such

( ) ( ) 2 sin( ). *ikx ikx*

Substituting φ(x) in the Schrödinger equation one can obtain the following expression for

2 2 2 *k E*

The identity is verified if ka = nπ, with n=1, 2, 3, ... The parameter k, therefore, results quantized, and the separation between two consecutive allowed values is Δk=π/a. In terms

> 22 2 22 2 . 2 2 *k n E m ma*

*ma*

The expression above defines the minimum energy possessed by a quantized system and is termed "point-zero energy" and constitutes a fascinating manifestation of the Heisenberg

2

*x a*

2 2 2

*m m ma*

since a represents the potential well dimension and Δx ≤ a/2. In terms of averaged kinetic

( ) . 2 2 2 *k k*

This expression gives the theoretical lower limit of the possible kinetic energy value for a quantum particle confined in a box (one-dimensional, in this case) having size a. In the

The lowest energy for the electron, called Ground State energy, is obtained for n = 1,

*E*

2 *p*

*p p E E*

reality we find an actual value about π2 ≈ 10 times larger.

*x A e e iA kx* (3)

*m* . (4)

( ) 2 sin( ) 0. *a iA ka* (5)

(6)

(7)

*x p* (8)

(9)

2

(10)

The condition φ(a) =0 produces the following identity:

that:

the electron energy:

of energy:

follows

energy:

Uncertainty Principle. In fact, from

The wavefunctions look like the standing waves on a string. Figure 1 plots the energies of the allowed states and the relative wavefunctions. They have increasing number of nodes with increasing energy according to the fact that more nodes mean shorter wavelength and higher momentum (i.e., energy).

#### **2.2 Classification of quantum confined systems**

In general, all the quantum confined systems can be classified on the basis of the number of dimensions along which the motion of electrons is coerced. The usual terminology refers to as 1-, 2-, 3-D confinement, in which the more evident effect consists in the modulation of the density of states function and a restriction of the allowed energies. In order to better understand what happens to a system when its dimensions start to shrink, one needs to begin from a non-confined structure, such as a bulk material. Here, one can assume that N electrons are not bound to individual atoms such that they can be considered "free" to move in three directions. If we suppose that, as a first approximation, the electron-electron interactions and the crystal potential are negligible (free electron gas model) (Pines, 1963) we can write the kinetic energy of an electron moving in the solid with velocity v=(v ,v ,v ) x y z :

$$E = \frac{1}{2}m\vec{v}^2 = \frac{1}{2}m(\upsilon\_x{}^2 + \upsilon\_y{}^2 + \upsilon\_z{}^2). \tag{12}$$

The corresponding wavevector is derived from the relation

$$
\vec{p} = m\vec{v} = \hbar \vec{k}\tag{13}
$$

and the corresponding wavelength λ=2π/ k .

On the wavefunction point of view the condition of infinite solid is expressed by imposing the so called boundary conditions. They consist in the continuity of the wavefunction φ(x,y,z) at the border of the real finite solid of dimensions dx, dy, dz:

$$\begin{cases} \varphi(\mathbf{x}, \mathbf{y}, \mathbf{z}) = \varphi(\mathbf{x} + d\_{\mathbf{x}'} \mathbf{y}, \mathbf{z}) \\ \varphi(\mathbf{x}, \mathbf{y}, \mathbf{z}) = \varphi(\mathbf{x}, \mathbf{y} + d\_{\mathbf{y}'} \mathbf{z}) \cdot \\ \varphi(\mathbf{x}, \mathbf{y}, \mathbf{z}) = \varphi(\mathbf{x}, \mathbf{y}, \mathbf{z} + d\_{\mathbf{z}}) \end{cases} \tag{14}$$

The solution of the 3-D Schrödinger equation gives a factored function, written as the product of three independent functions:

$$
\varphi(\mathbf{x}, y, z) = \varphi(\mathbf{x})\varphi(y)\varphi(z) = A\exp(ik\_x \mathbf{x})\exp(ik\_y y)\exp(ik\_z z) \tag{15}
$$

Optical Properties of Spherical Colloidal Nanocrystals 97

can consider a continuous distribution in the x-y plane and an ensemble of infinite states equidistant one to each other. The number of states with k in the interval k k Δk is now

> ( ) ( ) *d dN k Dk k*

This expression means that in the motion plane the density of the states is constant. Obviously, this holds for a fixed value of kz. By changing kz results in an increasing of the

In a one-dimensional system another direction of confinement is introduced making the structure resembling to a long wire. In the long dimension the DOS function becomes

> ( ) ( ) *d dN k Dk C*

() () *d* ( ) 1 / *dN E dN k dk D E E*

As in the 2-D case, an energy quantization is present and the function D1d (E) follows a sawtooth-like function. In particular, it is characterized by a quasi-continuous distribution of

When the carriers are confined in all the three spatial dimensions the system undergoes a triple quantization. As a consequence the wavenumbers of the allowed states (kx, ky, kz) are discrete and can be represented by a point in the k-space. In terms of energy, the Density of States collapses into a series of discrete values assuming a Dirac's Delta function, as shown

On the optical point of view, the main quantity considered in studying the properties of semiconductor materials (either bulk or confined systems) is their energy gap. It represents the energy spacing between the last occupied and the first unoccupied electronic state at the

Here, E0 represents the energy gap at 0 K of the bulk semiconductor, Eq is the contribution of the quantization effect and EC accounts for the Coulomb interaction between the e-h pairs present in the semiconductor. The relative contribution of the last two terms determines the degree of confinement that a system undergoes. To this purpose, it is useful to introduce the

() () *d* ( ) / *dN E dN k dk D E E EC*

2

1

states (the hyperbolas in figure 2), but delimited by a series of singularities.

direction). On the Density of States point of view, we

*dk* (19)

*dk* (21)

*dE dk dE* . (22)

*E EEE g q* 0 *C* . (23)

*dE dk dE* . (20)

Δk =z z π/d (dz being the size in the z

In terms of energy:

from which

proportional to k Δk and the consequent DOS function

2

density D2d (E) which assumes now a step-like function.

1

in figure 2, resembling to an atomic-like distribution.

ideal temperature of 0 K and is defined as

In the argument of the exponential functions, kx,y,z is such that ±Δk =±n2 x,y,z π/dx,y,z with n integer. In a bulk (having d much larger than the electronic wavefunction), this condition tells us that all the values of allowed k are contained in a sphere in the k-space with a quasicontinuous distribution of states. At this point, it is useful to introduce the concept of Density of States (DOS) function D (k) 3d , intended as the number of states for unitary interval of wavenumbers. The electrons in a solid having a wavenumber k included in the interval between k and k+Δk belong to the function D (k) 3d Δk . The total number of electrons contained in the sphere (having a maximum wavenumber kmax) is

$$N = \int\_{o}^{k\_{\text{max}}} D\_{3d}(k) dk \tag{16}$$

Since the volume of the sphere is proportional to k3 the number ΔN (k) of electrons inside the interval k+Δk is proportional to k2Δk. Therefore

$$D\_{3d}(k) = \frac{dN(k)}{dk} \propto k^2 \tag{17}$$

Now, it is possible to give the expression of the number of states in a unitary interval of energy D3d (E). Since 2 E(k) k , thus k E , dk/dE 1/ E . It follows

$$D\_{3d}(E) = \frac{dN(E)}{dE} = \frac{dN(k)}{dk}\frac{dk}{dE} \propto E \ne \sqrt{E} = \sqrt{E} \tag{18}$$

In figure 2 we can see the situation in terms of Energy and Density of States.

Fig. 2. Evolution of the "Density of States" (DOS) function by varying the degree of confinement. Reprinted from Physics Reports, 501, Krahne, R.; Morello, G.; Figuerola, A.; George, C.; Deka, S. & Manna, L., Physical properties of elongated inorganic nanoparticles, 75-221, Copyright (2012), with permission from Elsevier.

The next step is to consider a system in which just a spatial dimension is reduced down to few nanometres, so that the electrons can freely move on a plan giving rise to the so called "2-D electron gas" (Davies, 1998). The quantization acts in the shrunk dimension (let us say the direction z ) along which the wavenumber assumes discrete values, k =n zzz Δk where

Δk =z z π/d (dz being the size in the z direction). On the Density of States point of view, we can consider a continuous distribution in the x-y plane and an ensemble of infinite states equidistant one to each other. The number of states with k in the interval k k Δk is now proportional to k Δk and the consequent DOS function

$$D\_{2d}(k) = \frac{dN(k)}{dk} \propto k \tag{19}$$

In terms of energy:

96 Fingerprints in the Optical and Transport Properties of Quantum Dots

In the argument of the exponential functions, kx,y,z is such that ±Δk =±n2 x,y,z π/dx,y,z with n integer. In a bulk (having d much larger than the electronic wavefunction), this condition tells us that all the values of allowed k are contained in a sphere in the k-space with a quasicontinuous distribution of states. At this point, it is useful to introduce the concept of Density of States (DOS) function D (k) 3d , intended as the number of states for unitary interval of wavenumbers. The electrons in a solid having a wavenumber k included in the interval between k and k+Δk belong to the function D (k) 3d Δk . The total number of electrons

max

*o*

( ) ( ) *d dN k Dk k*

() () *d*( ) / *dN E dN k dk D E EE E*

Fig. 2. Evolution of the "Density of States" (DOS) function by varying the degree of confinement. Reprinted from Physics Reports, 501, Krahne, R.; Morello, G.; Figuerola, A.; George, C.; Deka, S. & Manna, L., Physical properties of elongated inorganic nanoparticles,

The next step is to consider a system in which just a spatial dimension is reduced down to few nanometres, so that the electrons can freely move on a plan giving rise to the so called "2-D electron gas" (Davies, 1998). The quantization acts in the shrunk dimension (let us say

) along which the wavenumber assumes discrete values, k =n zzz Δk where

*k*

3

energy D3d (E). Since 2 E(k) k , thus k E , dk/dE 1/ E . It follows

In figure 2 we can see the situation in terms of Energy and Density of States.

3 ( )

2

*N D k dk* (16)

*dk* (17)

*dE dk dE* (18)

*d*

Since the volume of the sphere is proportional to k3 the number ΔN (k) of electrons inside

Now, it is possible to give the expression of the number of states in a unitary interval of

contained in the sphere (having a maximum wavenumber kmax) is

the interval k+Δk is proportional to k2Δk. Therefore

3

75-221, Copyright (2012), with permission from Elsevier.

the direction z

$$D\_{2d}(E) = \frac{dN(E)}{dE} = \frac{dN(k)}{dk}\frac{dk}{dE} \propto \sqrt{E} \ne \sqrt{E} \ll \mathcal{C} \,. \tag{20}$$

This expression means that in the motion plane the density of the states is constant. Obviously, this holds for a fixed value of kz. By changing kz results in an increasing of the density D2d (E) which assumes now a step-like function.

In a one-dimensional system another direction of confinement is introduced making the structure resembling to a long wire. In the long dimension the DOS function becomes

$$D\_{1d}(k) = \frac{dN(k)}{dk} \propto \mathbb{C} \tag{21}$$

from which

$$D\_{1d}(E) = \frac{dN(E)}{dE} = \frac{dN(k)}{dk}\frac{dk}{dE} \propto 1 \,/\sqrt{E} \,. \tag{22}$$

As in the 2-D case, an energy quantization is present and the function D1d (E) follows a sawtooth-like function. In particular, it is characterized by a quasi-continuous distribution of states (the hyperbolas in figure 2), but delimited by a series of singularities.

When the carriers are confined in all the three spatial dimensions the system undergoes a triple quantization. As a consequence the wavenumbers of the allowed states (kx, ky, kz) are discrete and can be represented by a point in the k-space. In terms of energy, the Density of States collapses into a series of discrete values assuming a Dirac's Delta function, as shown in figure 2, resembling to an atomic-like distribution.

On the optical point of view, the main quantity considered in studying the properties of semiconductor materials (either bulk or confined systems) is their energy gap. It represents the energy spacing between the last occupied and the first unoccupied electronic state at the ideal temperature of 0 K and is defined as

$$E\_{\mathcal{R}} = E\_0 + E\_q + E\_{\mathcal{C}} \,. \tag{23}$$

Here, E0 represents the energy gap at 0 K of the bulk semiconductor, Eq is the contribution of the quantization effect and EC accounts for the Coulomb interaction between the e-h pairs present in the semiconductor. The relative contribution of the last two terms determines the degree of confinement that a system undergoes. To this purpose, it is useful to introduce the

Optical Properties of Spherical Colloidal Nanocrystals 99

CdSe and CdTe nanocrystals probably represent the most investigated nanostructures, for what concerns both their optical and their electronic properties. About the electronic structure of the lowest energy state, we will highlight some of the pioneering works which have represented important milestones in the field, namely the theoretical and experimental studies of Efros et al. (Efros et al., 1996). The concepts discussed here can be considered as of general validity for nanocrystals having hexagonal or cubic crystallographic structure and

In our discussion, we consider a nanocrystal with dimensions that are much larger than its lattice constant, such that the effective mass approximation (Efros et al., 1996) is applicable. This condition is practically fulfilled in all cases, since the nanocrystal diameter is hardly smaller than 2–3 nm. The notation used to name the quantum states of a nanocrystal closely follows that of an atomic system, consistently to the energy expression previously determined. Therefore, we define the total angular momentum J=(L+S) as the sum of the total orbital angular momentum L and the multiplicity term S, i.e. the electron spin, and the relative momentum projections: j, l, and s. The electron ground state has s-symmetry and presents a double degeneracy, which is due exclusively to the spin momentum. Thus J=0+1/2, its projections are j=±1/2, and the state is conventionally indicated as 1Se. On the other hand, the first hole level, having a p-symmetry, is fourfold degenerate, having J=1+1/2=3/2 (j=3/2, 1/2, -1/2, -3/2), and is named 1S3/2. The composition of the two

**2.3 Exciton fine structure of wurtzite CdSe and cubic CdTe nanocrystals**

ground states yields the eightfold degenerate exciton ground state 1S3/21Se.

Fig. 3. (a) The dependence of the hole ground state function φ(β) on the light to heavy hole effective mass ratio β; (b) v(β) associated with hole level splitting due to hexagonal lattice structure; (c) u(β) associated with hole level splitting due to crystal shape asymmetry; (d) χ(β) associated with exciton splitting due to the electron-hole exchange interaction. Reprinted figure with permission from Efros, Al. L.; Rosen, M.; Kuno, M.; Nirmal, M.; Norris, D. J.& Bawendi, M. Physical Review B, 54, 4843-4856, 1996. Copyright (2012) by the

American Physical Society.

are applicable to nanocrystals of a wide range of dimensions.

fundamental concept of the "exciton". An exciton is a bound system composed of an electron and a hole experiencing a mutual Coulomb interaction. An important parameter of the exciton is the so called Bohr radius. It is a characteristic quantity of each bulk material and represents the natural extension of the e-h pair when the carriers are free to orbit one around to each other. A parameter linked to the exciton and to its Bohr radius aB is the binding energy of the exciton Eb:

$$E\_b = \frac{\hbar^2}{2\mu a\_B^{-2}}\,. \tag{24}$$

where μ is the reduced mass of the exciton, defined as e h 11 1 = + μ m m , me and mh being the

effective masses of electron and hole, respectively. This energy assumes always negative values (being a binding energy) and represents the energy needed to ionize the exciton, that is to sustain the electron and the hole as separate entities. The Bohr radius can be expressed in terms of some physical parameters, such as the dielectric constant of the semiconductor ε, the reduced mass of the exciton μ and the electron charge e:

$$a\_{\mathcal{B}} = \frac{\hbar^2 \mathcal{E}}{\mu \varepsilon^2}.\tag{25}$$

Such quantity is different for each material and particle (i.e. electrons, holes, excitons, etc.) and ranges from a few nm to some tens of nm. Since the confinement acts as a constrainer for the e-h wavefunction extension, we can say that each material undergoes proper quantization effects, depending on its size and Bohr radius. In few words, the degree of confinement of a system expresses a measure of how much the real dimensions of the material affect the motion and the lifetime of an exciton in such material. Indeed, comparing the Bohr radius and the real size of a nanoparticle makes possible to define three different regimes of quantization: weak, intermediate and strong confinement. Let ae, ah and aex be the Bohr radii of electron, hole and exciton, respectively; if a is the size of the system considered (for instance the radius of a spherical nanoparticle), when a is smaller than ae, ah and aex the system is in the strong confinement regime since both the electron and the hole strongly feel the boundary of the nanocrystal. In the weak confinement regime, the nanocrystal size is larger than the electron and hole Bohr radius but smaller than the exciton Bohr radius. If a falls in between ae and ah, then the nanoparticle experiences the intermediate confinement effect. It is important to note, this point, that the degree of confinement featuring a semiconductor nanocrystal depends on the particular material constituting itself, since all the playing quantities are characteristic of each component. Therefore, nanocrystals of different materials but having the same size undergo a different degree of confinement. As a few examples we can consider three materials: CuCl, CdSe, InAs with radius of about 6 nm; their Bohr radii are 0.68 nm (Ohmura & Nakamura, 1999), 5.7 nm (Millo et el., 2004) and 34 nm (Kong et al., 2006), respectively. As a consequence, the effect of the quantum confinement effect is different in each nanocrystal: CuCl experiences a weak confinement, CdSe an intermediate one and InAs undergoes a strong confinement regime.

fundamental concept of the "exciton". An exciton is a bound system composed of an electron and a hole experiencing a mutual Coulomb interaction. An important parameter of the exciton is the so called Bohr radius. It is a characteristic quantity of each bulk material and represents the natural extension of the e-h pair when the carriers are free to orbit one around to each other. A parameter linked to the exciton and to its Bohr radius aB is the

> 2 2 2 *b B*

*a* . (24)

, me and mh being the

e h

. (25)

11 1 = + μ m m

effective masses of electron and hole, respectively. This energy assumes always negative values (being a binding energy) and represents the energy needed to ionize the exciton, that is to sustain the electron and the hole as separate entities. The Bohr radius can be expressed in terms of some physical parameters, such as the dielectric constant of the semiconductor ε,

> 2 *B* 2 *a e*

Such quantity is different for each material and particle (i.e. electrons, holes, excitons, etc.) and ranges from a few nm to some tens of nm. Since the confinement acts as a constrainer for the e-h wavefunction extension, we can say that each material undergoes proper quantization effects, depending on its size and Bohr radius. In few words, the degree of confinement of a system expresses a measure of how much the real dimensions of the material affect the motion and the lifetime of an exciton in such material. Indeed, comparing the Bohr radius and the real size of a nanoparticle makes possible to define three different regimes of quantization: weak, intermediate and strong confinement. Let ae, ah and aex be the Bohr radii of electron, hole and exciton, respectively; if a is the size of the system considered (for instance the radius of a spherical nanoparticle), when a is smaller than ae, ah and aex the system is in the strong confinement regime since both the electron and the hole strongly feel the boundary of the nanocrystal. In the weak confinement regime, the nanocrystal size is larger than the electron and hole Bohr radius but smaller than the exciton Bohr radius. If a falls in between ae and ah, then the nanoparticle experiences the intermediate confinement effect. It is important to note, this point, that the degree of confinement featuring a semiconductor nanocrystal depends on the particular material constituting itself, since all the playing quantities are characteristic of each component. Therefore, nanocrystals of different materials but having the same size undergo a different degree of confinement. As a few examples we can consider three materials: CuCl, CdSe, InAs with radius of about 6 nm; their Bohr radii are 0.68 nm (Ohmura & Nakamura, 1999), 5.7 nm (Millo et el., 2004) and 34 nm (Kong et al., 2006), respectively. As a consequence, the effect of the quantum confinement effect is different in each nanocrystal: CuCl experiences a weak confinement, CdSe an intermediate one and InAs

*E*

where μ is the reduced mass of the exciton, defined as

the reduced mass of the exciton μ and the electron charge e:

undergoes a strong confinement regime.

binding energy of the exciton Eb:

#### **2.3 Exciton fine structure of wurtzite CdSe and cubic CdTe nanocrystals**

CdSe and CdTe nanocrystals probably represent the most investigated nanostructures, for what concerns both their optical and their electronic properties. About the electronic structure of the lowest energy state, we will highlight some of the pioneering works which have represented important milestones in the field, namely the theoretical and experimental studies of Efros et al. (Efros et al., 1996). The concepts discussed here can be considered as of general validity for nanocrystals having hexagonal or cubic crystallographic structure and are applicable to nanocrystals of a wide range of dimensions.

In our discussion, we consider a nanocrystal with dimensions that are much larger than its lattice constant, such that the effective mass approximation (Efros et al., 1996) is applicable. This condition is practically fulfilled in all cases, since the nanocrystal diameter is hardly smaller than 2–3 nm. The notation used to name the quantum states of a nanocrystal closely follows that of an atomic system, consistently to the energy expression previously determined. Therefore, we define the total angular momentum J=(L+S) as the sum of the total orbital angular momentum L and the multiplicity term S, i.e. the electron spin, and the relative momentum projections: j, l, and s. The electron ground state has s-symmetry and presents a double degeneracy, which is due exclusively to the spin momentum. Thus J=0+1/2, its projections are j=±1/2, and the state is conventionally indicated as 1Se. On the other hand, the first hole level, having a p-symmetry, is fourfold degenerate, having J=1+1/2=3/2 (j=3/2, 1/2, -1/2, -3/2), and is named 1S3/2. The composition of the two ground states yields the eightfold degenerate exciton ground state 1S3/21Se.

Fig. 3. (a) The dependence of the hole ground state function φ(β) on the light to heavy hole effective mass ratio β; (b) v(β) associated with hole level splitting due to hexagonal lattice structure; (c) u(β) associated with hole level splitting due to crystal shape asymmetry; (d) χ(β) associated with exciton splitting due to the electron-hole exchange interaction. Reprinted figure with permission from Efros, Al. L.; Rosen, M.; Kuno, M.; Nirmal, M.; Norris, D. J.& Bawendi, M. Physical Review B, 54, 4843-4856, 1996. Copyright (2012) by the American Physical Society.

Optical Properties of Spherical Colloidal Nanocrystals 101

nanocrystals having a series of possible shapes, as depicted in figure 4 (Efros et al., 1996). The final configuration of the fine structure is based on the definition of the projection f of the total angular momentum F. It assumes different values: one state with f=±2, two states with f=±1 (named Upper and Lower, depending on the branch they originate) and two others with f=0 (Upper and Lower). Three of them are optically active, namely the states 0U, ±1U and ±1L, and the remaining ±2 and 0L states are passive. The ±2 state is optically forbidden because of the restrictions about the angular momentum conservation for photons (which cannot have an angular momentum ±2, for example). The 0L has zero optical transition probability because of an interference phenomenon between the two indistinguishable states with zero angular momentum (Efros & Rosen, 2000), due to the influence of the electron–hole exchange interaction. The shape of the nanocrystal plays the significant role on defining which of the above states represents the exciton ground state. For perfectly spherical nanocrystals the ±2 is the ground state, whereas in prolate nanocrystals an inversion of the ±2 with the 0L state can occur, because the state ±2 originates from the hole state with |j|=3/2, whilst 0L arises from the state |j|=1/2. When the conditions for the sign change of Δt are met, the ground state is inverted. A natural generalization of this concept can be found in the electronic structure of elongated nanocrystals (also named Nanorods (Krahne et al., 2011)) which can be approximated by axially symmetric prolate ellipsoids with ellipticity ε defined as ε = (2aB/b)-1 (here, the long axis c can be replaced by the Bohr radius), with b being the ellipsoid diameter and aB the Bohr radius. In the case of strong lateral confinement (b<2aB), the ellipsoids are subject to a possible inversion of the ground state between ±2 and 0L. This can happen because Δ2 becomes increasingly important in the strong confinement regime, and at some point it would cause the light hole state with j=±1/2 to become the hole ground state. The coupling with the electron state 1Se yields a fourfold degenerate state, with angular momentum 0 (two states) and ±1. The hole state with j=±3/2 yields the second doubly degenerate state with momentum ±1 and ±2. In practice, the new lowest exciton level would be the state 0L and the exciton fine structure resembles that of figure 4 c. For what concerns nanocrystals having a cubic structure, we will refer to another case study represented by CdTe nanocrystals. Efros et al. (Efros et al., 1996) calculated the size dependence of the band-edge splitting, showed in figure 5 for different shapes. In spherical nanocrystals the e-h exchange interaction split the exciton ground state into two states. The state at lowest energy is fivefold degenerate, presents total angular momentum F=2 and results dark. The higher energy state is threefold degenerate, has a total angular momentum F=1 and is bright. Contrary to the wurtzite case, the crystal field does not act as splitter, and just the shape anisotropy contributes to determine

the order of the excitonic states in the real system (Efros et al., 1996).

On the experimental point of view, the fine structure of CdSe nanocrystals has been investigated by a number of experiments, more than CdTe one which presents a lot of degeneracy. The main difficulty in studying the fine structure is given by the size distribution of a sample of colloidal nanocrystals leading to an inhomogeneous broadening of the optical spectra which hides the fine distribution of the states. On the other hand, single nanoparticle experiments can provide information only ideally, since several effects contribute to destroy the advantages of spectral narrowing, namely intermittent emission (blinking) (Schlegel et al., 2002), spectral diffusion (Empedocles & Bawendi, 1999) and possible photodegradation (Wang et al., 2003). Fluorescence line narrowing and photoluminescence excitation (Norris et al., 1996) experiments are the most useful steadystate methods to access a subset of the nanocrystals ensemble, whereas the cross polarized,

The degeneracy of the ground state is lifted by various effects, such the internal crystal field, shape effects and the electron–hole exchange interaction. The first effect arises from an intrinsic property of semiconductors having hexagonal lattice structure (like CdSe) and therefore manifests itself in both bulk and nanoscale materials. The second effect takes into account the deviation from the ideal spherical shape of nanocrystals, while the third effect accounts for mixing of electron and hole spins. The first two effects can be grouped together, as they arise from the intrinsic asymmetry of the material/nanocrystal. The intrinsic crystal field produces a first splitting of the valence band, i.e. the lowest hole state, in the so called Kramers doublet, which consists of two doubly degenerate states with j=±1/2 and j=±3/2 (Efros et al., 1996). Let us define a parameter β as the ratio between the mass of the light hole mlh and the mass of the heavy hole mhh (hence β = mlh/mhh). The energetic splitting due to the intrinsic crystal field is then expressed as (Efros et al., 1996):

$$
\Delta\_1 = \Delta\_{\mathbb{C}\mathcal{F}} \nu(\mathcal{J}) \tag{26}
$$

where ΔCF is a parameter related to the crystal field (CF) splitting in a crystal with hexagonal structure, contributing to determine the hole ground state with |j|=3/2, while ν (β) is a function which is unique for each material (see (Efros et al., 1996) for details). It is worth to stress that Δ1 does not depend on the size of the nanocrystal. Moreover, since ν (β) is always positive (see figure 3 b), the lowest hole level is fixed with the heavy hole state with |j|=3/2. In order to take the shape anisotropy into account, we can model a nanocrystal as an axially symmetric ellipsoidal particle (i.e. an ellipsoid with principal axes a = b < c), and define the ratio of the major to minor axes as c/b = 1+ε, ε being the ellipticity. The induced splitting in this case is:

$$
\Delta\_2 = 2\mathfrak{u}(\mathcal{J}) E\_{3/2}(\mathcal{J}) \varepsilon \tag{27}
$$

Here, u (β) is a dimensionless function associated with the hole level splitting due to the crystal shape (for details see (Efros et al., 1996), figure 3 c) and E3/2 (β) is the hole ground state energy which can be written as:

$$E\_{3/2}(\beta) = \frac{\hbar^2 \sigma^2(\beta)}{2m\_{\rm hlr}a^2} \tag{28}$$

where φ2(β) is a term related to the spherical Bessel functions and a is related to the nanocrystal size in the sense that for quasi-spherical nanocrystals a = (b2c)1/3.

Concerning Δ2, an important point is the trend in the function u (β), in particular for what concerns its sign. As shown in Figure 3 c, u (β) reverses its sign past a certain value of β, meaning that for some materials the shape anisotropy induces a negative splitting resulting in a possible inversion of the hole ground state between |j|=3/2 and |j|=1/2, since the global energy splitting is the sum of the single asymmetry contributions (Δt = Δ1 + Δ2). A negative Δ2 is found, for example, in elongated CdSe nanorods, for which β = 0.28, where a possible inversion would depend on the hole ground state energy and on the radius of rods (Krahne et al., 2011).

The exciton ground state results until now split in two fourfold degenerate excitonic states, having total angular momentum F=1 and F=2. The exchange interaction further contributes to an increase of the splitting of the remaining states, defining the fine structure for

The degeneracy of the ground state is lifted by various effects, such the internal crystal field, shape effects and the electron–hole exchange interaction. The first effect arises from an intrinsic property of semiconductors having hexagonal lattice structure (like CdSe) and therefore manifests itself in both bulk and nanoscale materials. The second effect takes into account the deviation from the ideal spherical shape of nanocrystals, while the third effect accounts for mixing of electron and hole spins. The first two effects can be grouped together, as they arise from the intrinsic asymmetry of the material/nanocrystal. The intrinsic crystal field produces a first splitting of the valence band, i.e. the lowest hole state, in the so called Kramers doublet, which consists of two doubly degenerate states with j=±1/2 and j=±3/2 (Efros et al., 1996). Let us define a parameter β as the ratio between the mass of the light hole mlh and the mass of the heavy hole mhh (hence β = mlh/mhh). The energetic splitting due to

> 1 ( ) *CF*

2 3/2 2( ) ( ) *u E*

Here, u (β) is a dimensionless function associated with the hole level splitting due to the crystal shape (for details see (Efros et al., 1996), figure 3 c) and E3/2 (β) is the hole ground

> 3/2 2 ( ) ( ) 2 *hh*

where φ2(β) is a term related to the spherical Bessel functions and a is related to the

Concerning Δ2, an important point is the trend in the function u (β), in particular for what concerns its sign. As shown in Figure 3 c, u (β) reverses its sign past a certain value of β, meaning that for some materials the shape anisotropy induces a negative splitting resulting in a possible inversion of the hole ground state between |j|=3/2 and |j|=1/2, since the global energy splitting is the sum of the single asymmetry contributions (Δt = Δ1 + Δ2). A negative Δ2 is found, for example, in elongated CdSe nanorods, for which β = 0.28, where a possible inversion would depend on the hole ground state energy and on the radius of rods

The exciton ground state results until now split in two fourfold degenerate excitonic states, having total angular momentum F=1 and F=2. The exchange interaction further contributes to an increase of the splitting of the remaining states, defining the fine structure for

*E*

nanocrystal size in the sense that for quasi-spherical nanocrystals a = (b2c)1/3.

2 2

*m a*

where ΔCF is a parameter related to the crystal field (CF) splitting in a crystal with hexagonal structure, contributing to determine the hole ground state with |j|=3/2, while ν (β) is a function which is unique for each material (see (Efros et al., 1996) for details). It is worth to stress that Δ1 does not depend on the size of the nanocrystal. Moreover, since ν (β) is always positive (see figure 3 b), the lowest hole level is fixed with the heavy hole state with |j|=3/2. In order to take the shape anisotropy into account, we can model a nanocrystal as an axially symmetric ellipsoidal particle (i.e. an ellipsoid with principal axes a = b < c), and define the ratio of the major to minor axes as c/b = 1+ε, ε being the ellipticity. The induced

(26)

(27)

(28)

the intrinsic crystal field is then expressed as (Efros et al., 1996):

splitting in this case is:

(Krahne et al., 2011).

state energy which can be written as:

nanocrystals having a series of possible shapes, as depicted in figure 4 (Efros et al., 1996). The final configuration of the fine structure is based on the definition of the projection f of the total angular momentum F. It assumes different values: one state with f=±2, two states with f=±1 (named Upper and Lower, depending on the branch they originate) and two others with f=0 (Upper and Lower). Three of them are optically active, namely the states 0U, ±1U and ±1L, and the remaining ±2 and 0L states are passive. The ±2 state is optically forbidden because of the restrictions about the angular momentum conservation for photons (which cannot have an angular momentum ±2, for example). The 0L has zero optical transition probability because of an interference phenomenon between the two indistinguishable states with zero angular momentum (Efros & Rosen, 2000), due to the influence of the electron–hole exchange interaction. The shape of the nanocrystal plays the significant role on defining which of the above states represents the exciton ground state. For perfectly spherical nanocrystals the ±2 is the ground state, whereas in prolate nanocrystals an inversion of the ±2 with the 0L state can occur, because the state ±2 originates from the hole state with |j|=3/2, whilst 0L arises from the state |j|=1/2. When the conditions for the sign change of Δt are met, the ground state is inverted. A natural generalization of this concept can be found in the electronic structure of elongated nanocrystals (also named Nanorods (Krahne et al., 2011)) which can be approximated by axially symmetric prolate ellipsoids with ellipticity ε defined as ε = (2aB/b)-1 (here, the long axis c can be replaced by the Bohr radius), with b being the ellipsoid diameter and aB the Bohr radius. In the case of strong lateral confinement (b<2aB), the ellipsoids are subject to a possible inversion of the ground state between ±2 and 0L. This can happen because Δ2 becomes increasingly important in the strong confinement regime, and at some point it would cause the light hole state with j=±1/2 to become the hole ground state. The coupling with the electron state 1Se yields a fourfold degenerate state, with angular momentum 0 (two states) and ±1. The hole state with j=±3/2 yields the second doubly degenerate state with momentum ±1 and ±2. In practice, the new lowest exciton level would be the state 0L and the exciton fine structure resembles that of figure 4 c. For what concerns nanocrystals having a cubic structure, we will refer to another case study represented by CdTe nanocrystals. Efros et al. (Efros et al., 1996) calculated the size dependence of the band-edge splitting, showed in figure 5 for different shapes. In spherical nanocrystals the e-h exchange interaction split the exciton ground state into two states. The state at lowest energy is fivefold degenerate, presents total angular momentum F=2 and results dark. The higher energy state is threefold degenerate, has a total angular momentum F=1 and is bright. Contrary to the wurtzite case, the crystal field does not act as splitter, and just the shape anisotropy contributes to determine the order of the excitonic states in the real system (Efros et al., 1996).

On the experimental point of view, the fine structure of CdSe nanocrystals has been investigated by a number of experiments, more than CdTe one which presents a lot of degeneracy. The main difficulty in studying the fine structure is given by the size distribution of a sample of colloidal nanocrystals leading to an inhomogeneous broadening of the optical spectra which hides the fine distribution of the states. On the other hand, single nanoparticle experiments can provide information only ideally, since several effects contribute to destroy the advantages of spectral narrowing, namely intermittent emission (blinking) (Schlegel et al., 2002), spectral diffusion (Empedocles & Bawendi, 1999) and possible photodegradation (Wang et al., 2003). Fluorescence line narrowing and photoluminescence excitation (Norris et al., 1996) experiments are the most useful steadystate methods to access a subset of the nanocrystals ensemble, whereas the cross polarized,

Optical Properties of Spherical Colloidal Nanocrystals 103

heterodyne detected third-order transient grating (CPH-3TG) method (Kim et al., 2009) has been employed to probe ultrafast transient dynamics in the fine structure distribution. On the luminescence point of view, the main difficulty is to be able to detect the very fast signal arising from the higher energy state of the fine structure. On this regard, a few time resolved PL experiments have been reported in the literature, showing emission from higher states on the picoseconds time scale (Morello et al., 2007a; Moreels et al., 2011) and only by using a streak camera. Also in these experiments the inhomogeneous broadening contributes to hide some spectral features and the situation is further complicated by both the possible exchange among the states due to the unavoidable asymmetric shape (Moreels et al., 2011) and the intermixing between intrinsic and surface states (Califano et al., 2005; Morello et al., 2007). All these effects participate to the definition of the actual ground state inside the fine structure. Indeed, the ideal case of perfectly spherical and pure nanocrystals is often replaced by the real situation in which slightly asymmetric and defected nanocrystals are investigated. Therefore, it is not rare to assist to the inversion between bright and dark states (it is the case of most nanorods systems (Krahne et al., 2011) and some not perfectly spherical NCs) or to the mixing between dark and surface states leading to the partial brightening of the same (Califano et al., 2005). The role of surface states in terms of quantum

yield and lifetime will be treated in the last part of the chapter.

by means of phonon emission instead of radiative emission.

**3. Excitation and relaxation processes in colloidal nanocrystals**

After excitation of charge carriers, the system tends to restore the initial equilibrium state losing the excess energy. All the processes involved in this action constitute the whole of the relaxation processes of the system. Restoring of the initial equilibrium state can occur in several ways, and dealing with nanoparticles instead of bulk semiconductor introduces some complications that will be analysed in the following paragraphs. The most common energetic relaxation process is the heating of the material. It consists in the trigger of an emission of phonons (lattice vibrations), mediated by the electron-phonon interaction. This kind of interaction does not affect only the decay toward lower energy states (carrier cooling (Nozik, 2001)), but it can influence the promotion of charge carriers to higher states through the absorption of phonons inducing, in particular cases, the ejection of charges away from the material. This particular process, holding for both electrons and holes, is strongly dependent on the temperature and is often referred to as "thermal escape" (Valerini et al., 2005; Morello et al., 2007b). In the following we will give a complete description of thermal escape occurring in CdSe (Valerini et al., 2005) and CdTe nanocrystals (Morello et al., 2007b). Another phenomenon of electron-hole annihilation is due to carrier-carrier scattering, via the so called "Auger effect". It manifests when more than one e-h pair is present into the semiconductor and consists in the transfer of the energy owned by an e-h couple to a third particle (another electron, hole, or exciton). This relaxation pathway is strongly dependent on both the morphology of the semiconductor (shape and dimensions) and the level of charge carrier injection. The alternative way for exciton relaxation is the emission of photons that, in general, is less efficient than heating. The reason lies on the fact that the emission (likewise absorption) of photons is hindered by the selection rules of the optical transitions: the photons can involve states with angular momentum equal to 0, ±1, whereas the branches of phonons are distributed on the whole first Brillouin's region, i.e. they possess all the possible electron momenta, thus allowing for connection of states having different angular momentum. This condition is particularly favourable in taking place of intraband relaxation

Fig. 4. Size dependence of the exciton ground state of CdSe NCs. (a) spherical NCs; (b) oblate NCs; (c) prolate NCs; (d) NCs having a size dependent ellipticity as determined from Efros et al., 1996. Solid/dashed lines indicate optically active/passive levels. Reprinted figure with permission from Efros, Al. L.; Rosen, M.; Kuno, M.; Nirmal, M.; Norris, D. J.& Bawendi, M. Physical Review B, 54, 4843-4856, 1996. Copyright (2012) by the American Physical Society.

Fig. 5. The exciton fine structure of cubic CdTe NCs. (a) spherical NCs; (b) oblate NCs; (c) prolate dots NCs; Solid/dashed lines indicate optically active/passive levels. Reprinted figure with permission from Efros, Al. L.; Rosen, M.; Kuno, M.; Nirmal, M.; Norris, D. J.& Bawendi, M. Physical Review B, 54, 4843-4856, 1996. Copyright (2012) by the American Physical Society.

Fig. 4. Size dependence of the exciton ground state of CdSe NCs. (a) spherical NCs; (b) oblate NCs; (c) prolate NCs; (d) NCs having a size dependent ellipticity as determined from Efros et al., 1996. Solid/dashed lines indicate optically active/passive levels. Reprinted figure with permission from Efros, Al. L.; Rosen, M.; Kuno, M.; Nirmal, M.; Norris, D. J.& Bawendi, M. Physical Review B, 54, 4843-4856, 1996. Copyright (2012) by the American Physical Society.

Fig. 5. The exciton fine structure of cubic CdTe NCs. (a) spherical NCs; (b) oblate NCs; (c) prolate dots NCs; Solid/dashed lines indicate optically active/passive levels. Reprinted figure with permission from Efros, Al. L.; Rosen, M.; Kuno, M.; Nirmal, M.; Norris, D. J.& Bawendi, M. Physical Review B, 54, 4843-4856, 1996. Copyright (2012) by the American Physical Society. heterodyne detected third-order transient grating (CPH-3TG) method (Kim et al., 2009) has been employed to probe ultrafast transient dynamics in the fine structure distribution. On the luminescence point of view, the main difficulty is to be able to detect the very fast signal arising from the higher energy state of the fine structure. On this regard, a few time resolved PL experiments have been reported in the literature, showing emission from higher states on the picoseconds time scale (Morello et al., 2007a; Moreels et al., 2011) and only by using a streak camera. Also in these experiments the inhomogeneous broadening contributes to hide some spectral features and the situation is further complicated by both the possible exchange among the states due to the unavoidable asymmetric shape (Moreels et al., 2011) and the intermixing between intrinsic and surface states (Califano et al., 2005; Morello et al., 2007). All these effects participate to the definition of the actual ground state inside the fine structure. Indeed, the ideal case of perfectly spherical and pure nanocrystals is often replaced by the real situation in which slightly asymmetric and defected nanocrystals are investigated. Therefore, it is not rare to assist to the inversion between bright and dark states (it is the case of most nanorods systems (Krahne et al., 2011) and some not perfectly spherical NCs) or to the mixing between dark and surface states leading to the partial brightening of the same (Califano et al., 2005). The role of surface states in terms of quantum yield and lifetime will be treated in the last part of the chapter.

### **3. Excitation and relaxation processes in colloidal nanocrystals**

After excitation of charge carriers, the system tends to restore the initial equilibrium state losing the excess energy. All the processes involved in this action constitute the whole of the relaxation processes of the system. Restoring of the initial equilibrium state can occur in several ways, and dealing with nanoparticles instead of bulk semiconductor introduces some complications that will be analysed in the following paragraphs. The most common energetic relaxation process is the heating of the material. It consists in the trigger of an emission of phonons (lattice vibrations), mediated by the electron-phonon interaction. This kind of interaction does not affect only the decay toward lower energy states (carrier cooling (Nozik, 2001)), but it can influence the promotion of charge carriers to higher states through the absorption of phonons inducing, in particular cases, the ejection of charges away from the material. This particular process, holding for both electrons and holes, is strongly dependent on the temperature and is often referred to as "thermal escape" (Valerini et al., 2005; Morello et al., 2007b). In the following we will give a complete description of thermal escape occurring in CdSe (Valerini et al., 2005) and CdTe nanocrystals (Morello et al., 2007b). Another phenomenon of electron-hole annihilation is due to carrier-carrier scattering, via the so called "Auger effect". It manifests when more than one e-h pair is present into the semiconductor and consists in the transfer of the energy owned by an e-h couple to a third particle (another electron, hole, or exciton). This relaxation pathway is strongly dependent on both the morphology of the semiconductor (shape and dimensions) and the level of charge carrier injection. The alternative way for exciton relaxation is the emission of photons that, in general, is less efficient than heating. The reason lies on the fact that the emission (likewise absorption) of photons is hindered by the selection rules of the optical transitions: the photons can involve states with angular momentum equal to 0, ±1, whereas the branches of phonons are distributed on the whole first Brillouin's region, i.e. they possess all the possible electron momenta, thus allowing for connection of states having different angular momentum. This condition is particularly favourable in taking place of intraband relaxation by means of phonon emission instead of radiative emission.

Optical Properties of Spherical Colloidal Nanocrystals 105

core/shell QDs is thermal escape, assisted by multiple longitudinal optical (LO) phonons absorption (Valerini et al., 2005), while at low temperature evidence for carrier trapping at surface defects was found. Despite these results, the role and the chemical origin (Wang et al., 2003b) of the surface defect states in the radiative and non radiative relaxation in nanocrystals has not been clarified completely. The existence of surface states due to unpassivated dangling bonds has been invoked to explain anomalous red-shifted emission bands in colloidal nanocrystals (Lee et al., 2005). On the contrary, it has been shown that above-gap trap states (Bawendi et al., 1992) affect the ultrafast relaxation dynamics (Cretì et al., 2005) and the single nanoparticle PL spectra of CdSe quantum rods (Rothenberg et al., 2005) due to charge trapping and local electric field fluctuations. These effects are expected to be dependent on the chemical composition of the QDs, on the density of surface defects and on the nanocrystals size. In what follows we will review the main processes affecting the light emission properties of nanocrystals, namely exciton-phonon coupling, Auger

The main features that distinguish nanostructures from bulk materials originate from the localized character of the electron and hole wavefunction and the discrete nature of their optical transitions. In terms of carrier-phonon coupling, it essentially affects and defines the properties of homogeneous broadening of the PL signal. In the bulk, the broadening is mainly determined by the polar coupling of both electrons and holes to optical phonons (Liebler et al., 1985). The piezoelectric and deformation potential coupling of both carriers to acoustic phonons is usually not very important (Takagahara, 1996). This situation is different in nanostructures, where the local charge neutrality character of the exciton becomes predominant, producing an ideal null polar coupling of the exciton to optical phonons (Schmitt-Rink et al., 1987). This holds true for infinite barriers, such that the electron and hole wavefunctions are practically identical. In general, in real systems (finite barrier) just a decrease of the polar coupling with increasing barrier is expected (Muljarov & Zimmerman, 2007; Nomura & Kobayashi, 1992; Schmitt-Rink et al., 1987). On the other hand, since the deformation potential coupling is proportional to 1/R2 (R being the radius of a spherical dot), the coupling strength to acoustic phonons is increased as the dimensions are reduced below the Bohr radius (Gindele et al., 2000; Takagahara, 1996). Also, the temperature affects each of these contributions. The temperature dependence of the spectral

> 0 ( ) ( ) *T T NT*

where Γ0 is the inhomogeneous broadening, σ is the exciton-acoustic phonons coupling coefficient, γ is the coefficient accounting for the exciton-optical (LO) phonon coupling and