Electronic structure of CdSe nanocrystals

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Electronic structure of CdSe nanocrystals

To understand the electronic structure of CdSe nanocrystals, one can start from the electronic properties of the bulk material and then take into account the quantum size effect [32, 33]. The other way around, treating the nanocrystals as a big molecule made of thousands of atoms is also possible [34–36] but less intuitive in terms of how to un-derstand the arising energy spectrum. By quantum size effect, one means that if the size of the semiconductor structure becomes comparable to or smaller than the De Broglie wavelength of the charge carriers (electrons and holes) for this specific material, then the charges start to feel the finite size of the material: they are confined inside the material. The confinement changes dramatically the energy states of the carriers. Among quantum confined systems, CdSe nanocrystals are part of the quantum dots family. They confine the charge carriers in the three dimensions of space. In this case, the energy spectrum moves from a continuum of allowed states for the bulk material to an atomic-like behavior as the bulk bands become quantized. This discretization of the energy level happens only because of the finite size as the material remains structurally identical to the bulk crystal.

From bulk CdSe to quantum confined nanocrystals

Bulk CdSe band structure

Cadmium Selenide is a II-VI direct bandgap semiconductor. Dispersion relations for the valence band (VB) and conduction band (CB) have a maximum and minimum respec-tively in k “ 0 where they can be approximated by parabolic dispersions:
m˚h and m˚e are the hole and electron effective masses considering they behave as free particles with a parabolic dispersion. The effective mass then simply models the compli-cated periodic potential felt by the carrier inside the crystal lattice. Eg is the bandgap of the material (Eg “ 1 84 eV» 690 nm for CdSe [33]).
Cadmium (rK rs4d10 5s2 ) atoms share their 5s electrons with Selenium (rArs3d10 4s2 4p4 ). The conduction band is then made of the empty Cadmium 5s orbitals. It is a twofold degenerate band [33] because of the electrons spin. It is well described by the effective mass approximation (effective mass m˚e “ 0 11m0 , with m0 the free electron mass).
The valence band is constituted of the 4p orbitals of Selenium filled with the 5s Cadmium electrons. It has an orbital angular momentum lh of 1 as it is essentially made of the more energetic p orbitals. The valence band is sixfold degenerate at k “ 0 [33]. The sixfold degeneracy is partly lift by the spin-orbit coupling. To consider this effect, the good quantum number to study is the hole total angular momentum Jh “ lh ` sh, the sum of the orbital and spin contribution to the angular momentum. Jh is either equal to 1{2 or 3{2, this gives two distinct bands, Jh “ 1{2 is called the split-off-hole band (SO) and separated by ∆SO “ 0 42 eV at k “ 0 from the higher Jh “ 3{2 band. This last one is further split for k ‰ 0 into two sub-bands called the heavy-hole (hh) and light-hole (lh) sub-bands with different associated effective masses owing to different projections Jhz of Jh onto the z axis (|Jhz | “ 1{2 or 3{2). Furthermore, the asymmetry of the wurtzite structure induces an intrinsic crystal field that splits the heavy-hole and light-hole sub-bands by an amount of ∆int “ 23 meV. Under the effective mass approximation, effective masses can be attributed to each of the valence band, m˚so, m˚hh and m˚lh for the split-off, heavy-hole and light-hole bands respectively. Fig.I.2 illustrates the bulk CdSe band structure.


After absorption of a high energy photon by a semiconductor material, an exciton, i.e. bound electron-hole pair2 , is created and can be described as a hydrogen-like atom. The binding between the electron and the hole is ensured by the Coulomb interaction. A Bohr radius can be defined for this particle: rB “ ǫ m0 r0 , (I.1) with r0 the Bohr radius of the hydrogen atom, ǫ the material dielectric constant, m0 the electron rest mass and µ the reduced mass of an electron-hole pair. For CdSe material, the Bohr radius is rB « 5 6 nm.The energy associated with the Coulomb interaction evolves as 1{r, r being the distance between the two particles, while the confinement energy (as will be demonstrated in the following paragraphs) evolves as 1{a2 , with a the confinement radius (for a spherical nanocrystal a is the radius). One way to picture the fact that an energy can be ascribed to the confinement is that according to the Heisenberg uncertainty principle the momentum of a particle, so its energy, increases if its position becomes well defined. Different confinement regimes are defined by comparing the confinement size with the exciton Bohr radius:
‚ The weak confinement regime: rB ă a. The Coulomb interaction is the dominant interaction and the quantum confinement is treated as a perturbation. This is typically the case for self-assembled quantum dots owing to their large size.
‚ The strong confinement regime: rB ą a. The Coulomb interaction is treated as a perturbation while the confinement is fully taken into account. This is the case for the CdSe nanocrystals.
Using the effective mass approximation previously described, the electronic structure of CdSe nanocrystals can be calculated. We suppose that the parabolic band model for the valence bands and conduction band as above described for the bulk material is still valid in the case of a nanocrystal. This is the case if the nanocrystal diameter is much larger than the lattice constant. For CdSe crystal, the lattice constant is p “ 0 43 nm such that the approximation is meaningful for nanometer size crystals. Then the charge carriers are evolving as free particles with a given effective mass inside a spherical potential well as depicted on Fig.I.3 for the specific case of a CdSe/CdS dot-in rod. For the following calculations the quantum well where the electron and hole are confined will be considered as infinite, a valid hypothesis for the lowest energy level. The case of a finite well has been treated in literature [38] with the consequences of a decrease of the confinement energy and a slight spreading of the wave functions outside the well.
According to the Bloch’s theorem, the electron and hole wavefunctions inside a bulk crystal can be written as: Ψnkrp q “ unkrp q exppikr q, (I.2)
where unk is a function of the crystal lattice periodicity that gives the wavefunction variations on a lattice site, while the other term is a plane wave term, it accounts for a larger scale dependency. k is the wavevector associated to the particle and the index n is associated to a specific band (conduction or valence band). Inside a nanocrystal, an electron or a hole wavefunction in a given band can be written as a linear combination of Bloch functions: Ψrp q “ Cnk unkrp q exppikr q, (I.3)
with the expansion coefficients Cnk satisfying the limit conditions of the infinite well problem. In practice, the Bloch’s functions are weakly dependent on k because the nanocrystal size a is much larger than the lattice constant p as already stated. In this case unk « un0 and the wavefunction can be simplified: ψrp q is called the envelop function. It describes the wavefunction at the nanocrystal scale and satisfies the Schrödinger equation. The confinement effects are described by the envelop function.

Electron confinement

Solving the Schrödinger equation for the envelop function ψrp q of an electron inside an infinite potential well yields the following energies [32, 33]: Enl “ nl , (I.5) e 2m˚e with knl the quantized values of k due to the finite size of the system. The envelop function solutions are hydrogen-like orbitals with n the radial quantum number, l and m the angular quantum numbers. By analogy with the atomic case, the electron state is written nLe and the orbitals L are called S, P . . . .The first quantized electronic state is then called 1Se and as the following energy: Ee1S “ ~π2 , (I.6) with k10 “ π{a. It is twofold degenerate with respect to the spin projection, with a spherical symmetry (1S state).

Hole confinement

The solutions for a hole in an infinite potential well are less straightforward to derive because of the complex valence band structure of bulk CdSe described above. The Hamil-tonian of the problem is called the Luttinger Hamiltonian [33,38,39], it takes into account the sixfold degenerate valence bands and therefore includes the interactions and mixing between these bands necessary to understand the experimental results [40].
Each valence sub-band does not produce its own ladder of hole states, but the hole states come from a mixture of the different valence sub-bands (the three ladders are cou-pled to each other). Without entering into details, the main result of these considerations is that one should take into account two quantum numbers for this problem: the envelop function orbital angular momentum L coming from the confined particle and the total angular momentum Jh from the underlying atomic basis forming the 3 hole sub-bands described in Fig.I.2. Therefore F “ L ` Jh is the good quantum number considering the Luttinger Hamiltonian3 . For a given F , the hole state is a combination of different envelop orbitals L (more precisely L and L ` 2, this is called “S-D” mixing) and the 3 sub-bands angular moment Jh (“valence band mixture”).
The hole energy level are written as follow: nLF . We just briefly introduce the first hole level which has a total angular momentum F “ 3{2, it is fourfold degenerate with the corresponding projections Fz “ p´3{2, ´1{2, 1{2, 3{2q. Following the introduced notation, it is called 1S3{2 . A global notation that shows the underlying mixture is: pF, Jh, Lq “ p3{2, 3{2, 0q, p3{2, 3{2, 2q, p3{2, 1{2, 2q.
This energy state enters into the exciton ground state together with the 1Se state to form the 1S3{2 1Se exciton state that is essential for the understanding of the photolumi-nescence properties of CdSe nanocrystals. This point is detailed below in sections I.1.2 and I.2.3.

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Excited states

Fig.I.4 presents results obtained in reference [40] from ensemble measurements on different CdSe nanocrystals samples at cryogenic temperature (10 K).
Fig.I.4a (top) shows an absorption spectrum of a CdSe dots sample of radius a “ 2 8 nm together with the associated emission spectrum obtained from an excitation at 2 7 eV (467 nm), well above the CdSe bandgap. The absorption spectrum displays dis-crete absorption features corresponding to the quantized exciton lines as expected from the 3D confinement of the charges. A first strong absorption line is visible at 2 1 eV ( 600 nm) corresponding to the band-edge 1S3{2 1Se exciton, followed by a weaker tran-sition a 100 meV further in the blue. The various lines cannot be fully resolved because of homogeneous and inhomogeneous broadening. The inhomogeneous broadening comes from the size distribution of the sample and the homogeneous broadening comes from the coupling of the excitation with the phonon modes of the crystals still present at 10 K. One can notice that conversely to the absorption spectrum, the emission spectrum (dashed line) is characterized by a single peak slightly red-shifted compared to the first absorption peak. The shift between the first and the second absorption lines depends on the size of the nanocrystal but is always much larger than the thermal energy even at room temperature („ 25 meV), such that the emission comes only from the 1S3{2 1Se state. Even if excited well above this transition, the nanocrystal excitation always re-lax non-radiatively towards the band-edge exciton before emitting a photon (see section I.2.2). CdSe nanocrystals are thus very appealing light emitters as they have a relatively narrow emission line for a solid state system that is highly tunable with the size of the confinement. The shift between the first absorption peak and the emission peak (referred as Stokes shift) is due to the fine structure [32, 41] of the band edge 1S3{2 1Se state. This is explained in the next section I.1.2.
Fig.I.4a (bottom) is a photoluminescence excitation (PLE) spectrum from the same ensemble of nanocrystals. Photoluminescence excitation measurement consists in select-ing a spectrally narrow emission window within the inhomogenous emission feature while scanning the frequency of the excitation source. Because excited nanocrystals always relax to their first excited state before emission, the signal that is obtained while scan-ning the excitation wavelength gives absorption information about the narrow subset of nanocrystals emitting at the wavelength that the experimentalist is looking at. This tech-nique reduces the inhomogeneous broadening compared to a simple absorption spectrum measurement. The absorption spectrum is then much more detailed in this case. Fitting such spectrum for different samples of various dot radii, Fig.I.4b from reference [40], that presents the first exciton lines energies, is obtained.
The different transitions energy are compared with the first transition energy from the 1S3{2 1Se exciton. These measurements contain many information and confirm the complexity of the transitions due to the valence sub-bands structure. The reader should refer to the article by Norris et al. ( [40]) for the full details as we will only briefly discuss the figure. As expected, the splitting between the lines increases for smaller dot radii as the confinement is stronger. Very weak transitions that are poorly visible on the PLE measurements are denoted with crosses. The strongest transitions are the 1S3{2 1Se and 1P3{2 1Pe transitions. The oscillator strengths of the interband transitions are proportionals to the electric dipole matrix element P between the vacuum state |0y and the exciton state under consideration |ΨX y: P “ |x0|Ep |ΨX y|2 , (I.7) with E the electric field operator andp the transition dipole moment. In the strong confinement regime the charges are treated separately (Coulomb interaction is treated as a perturbation), P can be expressed by separating the Bloch and envelop function parts of the carriers wavefunctions: P “ |pcv |2 | ψhψed3 r|2 “ |pcv |2 Peh, (I.8) where pcv is the dipole matrix element for the bulk material and Peh the carriers envelope wavefunctions overlap. Therefore the oscillator strength of a transition depends on the carriers envelop overlap.
As it can be seen on the absorption and PLE curves on Fig.I.4a, it is not possible to distinguish discrete transitions 1 eV above the fundamental interband transition for several reasons:
‚ the bands are not parabolic anymore but concave implying that energy states are very close to each other,
‚ the hole density of states is large at higher energies [42],
‚ the energy states are broadened by their coupling to phonons and overlap [42].
This continuum-like absorption spectrum implies that nanocrystals can easily be excited by various sources. Details about absorption concerning our experiment are given further in section I.2.1.

Fine structure of the band-edge exciton

Photoluminescence of CdSe nanocrystals originates from the band-edge exciton only ow-ing to the fast non-radiative relaxations [43] from higher states. To fully understand the photoluminescence properties, it is necessary to consider the fine structure of the band-edge exciton [32]. The fine structure explains the polarization properties of the emission, the Stokes shift of the emission [44] whose origin was first wrongly attributed to the emission of a surface state [45] and also the very long exciton emission lifetime („ 1 µs) measured at cryogenic temperatures [45] below 2K .

Lifted degeneracy

The band-edge exciton is eightfold degenerate (twice with respect to the 1Se electron spin projection and four times with respect to the 1S3{2 hole total angular momentum projection Fz ) for spherical dots with zinc-blende structure. Three effects have to be considered to lift the degeneracy:
‚ the hexagonal crystal structure,
‚ the nonsphericity of nanocrystals,
‚ the electron-hole exchange interaction.
Hexagonal crystal structure When considering a wurtzite structure, the intrinsic asymmetry of this structure generates a crystal field that splits the heavy-hole light-holes sub-bands [46] at k “ 0 (see section I.1.1 and Fig.I.2). The 1S3{2 hole state is split in two twofold degenerate sub-levels with |Jhz | “ 3{2 and |Jhz | “ 1{2. The corresponding energy ∆int is independent of the crystal size and is equal to 23 meV for CdSe [46].
Shape anisotropy As already mentioned in section I, the wurtzite CdSe nanocrystals are not perfectly spherical but often slightly elliptical [4]. The ellipticity is characterized by the parameter µ: µ “ c{b ´ 1, (I.9) with c and b the crystals axis length. If c is greater than b, µ is positive and the crystal is prolate (elongated along the c axis), conversely it is said to be oblate if µ is negative. As the crystal field4 , it splits the fourfold degenerate hole state with respect to the heavy-hole and light-hole bands total angular momentum projections |Jhz |. The shape anisotropy splitting is denoted ∆sh. Besides depending on the ellipticity µ, its value depends also on the 1S3{2 ground state hole energy and thus on the size of the nanocrystal (1{a2 dependence with the nanocrystal radius).
We will call ∆ “ ∆sh ` ∆int the net-splitting of the hole state due to the crystal field and shape anisotropy5 . The net-splitting can become negative for sufficiently small prolate nanocrystals as the shape anisotropy contribution can be negative, this reverse the hole state order.
Electron-hole exchange interaction Finally, the eightfold degeneracy of the spher-ical dot band-edge exciton is also broken by the electron-hole exchange interaction [32]. This Coulomb interaction is strongly enhanced in nanocrystals, especially when reach-ing the strong confinement regime for which the electron and hole wavefunctions overlap considerably [47]. It is proportional to the inverse of the nanocrystal volume („ 1{a3 ).

Table of contents :

I CdSe/CdS colloidal nanocrystals 
I.1 Electronic structure of CdSe nanocrystals
I.1.1 From bulk CdSe to quantum confined nanocrystals
I.1.2 Fine structure of the band-edge exciton
I.2 Optical transitions and relaxation processes
I.2.1 Highly excited states and absorption
I.2.2 Non-radiative relaxation processes in nanocrystals
I.2.3 Quantum states and radiative relaxation
I.2.4 Quantum yields
I.3 Core/shell nanocrystals
I.3.1 Classification of core/shell heterostructures
I.3.2 CdSe/CdS nanocrystals: quasi-type II heterostructures and wavefunction engineering
I.3.3 Nanorods
I.3.4 CdSe/CdS dot-in-rods
I.4 Studied Samples
I.4.1 Samples geometry
I.4.2 Samples spectra
II Experimental setup and methods 
II.1 Setup description
II.1.1 Standard microscopy setup
II.1.2 Polarization microscopy
II.1.3 Defocused microscopy
II.1.4 Collection efficiency
II.2 Measurements and data processing
II.2.1 Sample preparation
II.2.2 Noise
II.2.3 Excitation power and saturation curves
II.2.4 Data acquisition and processing
IIICdSe/CdS dot-in-rods blinking statistics 
III.1 Blinking: a review
III.1.1 Two types of blinking
III.1.2 Reduced blinking
III.2 Dot-in-rods blinking statistics
III.2.1 Type A blinking
III.2.2 Blinking timescales and statistics
IV CdSe/CdS dot-in-rods photon statistics 
IV.1 Photon statistics theory
IV.1.1 A general approach to the photon statistics
IV.1.2 Photon statistics and nanocrystals
IV.2 Single photon emission: exciton and negative trion
IV.2.1 Saturation and exciton quantum yield
IV.2.2 Trion quantum yield and electron delocalization
IV.3 Two photon emission: biexciton and charged biexciton
IV.3.1 Dot-in-rod geometry and biexciton quantum yield
IV.3.2 Blinking and charged biexciton emission
IV.4 Measuring the photon statistics with an ICCD camera
V CdSe/CdS dot-in-rods emission polarization 
V.1 Dot-in-rods polarization: an overview
V.1.1 Ensemble measurement methods
V.1.2 Dielectric effects
V.1.3 Fine structure
V.1.4 Polarization and c axis
V.2 Polarization of the emission and geometrical parameters
V.2.1 Measurements procedure
V.2.2 Measurements
V.3 Discussion
V.3.1 Comparison with the literature
V.3.2 A model for the emission polarization
V.3.3 Simulations of the emission polarization
VI Coupling nanocrystals to devices: towards integrated nanophotonics 
VI.1 Coupling nanocrystals to ZnO nanowires
VI.1.1 System description
VI.1.2 Passive/Active excitation
VI.1.3 Losses and excitation efficiency
VI.1.4 Outlook
VI.2 Orientation of CdSe/CdS dot-in-rods using liquid crystals
VI.2.1 Orientation and positioning of nanoparticles
VI.2.2 The liquid crystal samples
VI.2.3 Polarization microscopy of dot-in-rods inside a liquid crystal
VI.2.4 Outlook
Conclusions and outlook


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