Semiconductor Bohr radius

As the radius of a semiconductor crystallite approaches the exciton-Bohr-radius its electronic properties begin to change, whereupon quantum size effects can be expected. The Bohr radius ub of an exciton is given by... 

Colloidal CdS particles 2-7 nm in diameter exhibit a blue shift in their absorption and luminescence characteristics due to quantum confinement effects [45,46]. It is known that particle size has a pronounced effect on semiconductor spectral properties when their size becomes comparable with that of an exciton. This so called quantum size effect occurs when R < as (R = particle radius, ub = Bohr radius see Chapter 4, coinciding with a gradual change in the energy bands of a semiconductor into a set of discrete electronic levels. The observation of a discrete excitonic transition in the absorption and luminescence spectra of such particles, so called Q-particles, requires samples of very narrow size distribution and well-defined crystal structure [47,48]. Semiconductor nanocrystals, or... 

Optically excited semiconductor nanostructures show effects of QC if at least one spatial dimension of the material becomes comparable to, or smaller than the characteristic length scale (the classical Bohr radius) of an e-h pair. Different regimes of QC have been defined which depend on the semiconductor nanocrystallite size R relative to that of the Bohr radius of the exciton an, the electron ae or the hole ah ... 

The total energies and wavefunctions of the Hamiltonian (1) have been calculated as the eigenvalues and eigenvectors of a Cl matrix. Full Cl has been used for all calculations of quasi-one-dimensional quantum dots and for quasi-two-dimensional quantum dots with N = 2, while multi-reference Cl has been used for quasi-two-dimensional quantum dots with N = 3 and 4. The results are presented in atomic units. They can be scaled by the effective Bohr radius of 9.79 nm and the effective Hartree energy of 11.9 meV for GaAs semiconductor quantum dots [25,26]. 

Lanthanides doped into nanocrystalline semiconductors have been the subject of numerous investigations in the past decades. If the size of a semiconductor particle is smaller than the Bohr radius of the excitons, the so-called quantum confinement occurs. As a result, the band gap of the semiconductor increases and discrete energy levels occur at the edges of the valence and conduction bands (Bol et al., 2002 Bras, 1986). These quantum size effects have stimulated extensive interest in both basic and applied research. 

Generally, quantum size effects are not expected in lanthanide-doped nanoinsulators such as oxides since the Bohr radius of the exciton in insulating oxides, like Y2O3 and Gd2C>3, is very small. By contrast, the exciton Bohr radius of semiconductors is larger (e.g., 2.5 nm for CdS) resulting in pronounced quantum confinement effects for nanoparticles of about 2.5 nm or smaller (Bol et al., 2002). Therefore, a possible influence of quantum size effects on the luminescence properties of lanthanide ions is expected in semiconductor nanocrystals. 

One of the most attractive features of colloidal semiconductor systems is the ability to control the mean particle size and size distribution by judicious choice of experimental conditions (such as reactant concentration, mixing regimen, reaction temperature, type of stabilizer, solvent composition, pH) during particle synthesis. Over the last decade and a half, innovative chemical [69], colloid chemical [69-72] and electrochemical [73-75] methods have been developed for the preparation of relatively monodispersed ultrasmall semiconductor particles. Such particles (typically <10 nm across [50, 59, 60]) are found to exhibit quantum effects when the particle radius becomes smaller than the Bohr radius of the first exciton state. Under this condition, the wave functions associated with photogenerated charge carriers within the particle (vide infra) are subject to extreme... 

Quantum dots are inorganic semiconductor nanocrystals that possess physical dimensions smaller than the exciton Bohr radius, giving rise to the unique phenom-... 

In the semiconductors of greater polarity, the dielectric constants are smaller and the effective masses larger, and the same evaluation leads to 0.07 eV in zinc selcnidc, for example many of the impurity states can be occupied at room temperature. As the energy of the impurity states becomes deeper, the effective Bohr radius becomes smaller and the use of the effective mass approximation becomes suspect the error leads to an underestimation of the binding energy. Thus, in semiconductors of greatest polarity- and certainly in ionic crystals— impurity states can become very important and arc then best understood in atomic terms. We will return to this topic in Chapter 14, in the discussion of ionic crystals. 

As you may recall from Chapter 4, when an electron is promoted from the valence to conduction bands, an electron-hole pair known as an exciton is created in the bulk lattice. The physical separation between the electron and hole is referred to as the exciton Bohr radius (re) that varies depending on the semiconductor composition. In a bulk semiconductor crystal, re is significantly smaller than the overall size of the crystal hence, the exciton is free to migrate throughout the lattice. However, in a quantum dot, re is of the same order of magnitude as the diameter (D) of the nanocrystal, giving rise to quantum confinement of the exciton. Empirically, this translates to the strongest exciton confinement when D < 2r. ... 

For the purposes of this article we will limit our discussion to particles defined by a minimum of two dimensions less than 100 nm but usually with 2-dimenions less than 10 nm. Current interest in these materials can principally be traced to work by Luis Brus in the mid-1980s in which he pointed out that the band gap of a simple direct band gap semiconductor such as CdS should be dependent on its size once its dimensions were smaller than the Bohr radius [10]. Experimental work confirmed this suggestion. Initial samples were prepared by low temperature... 

Size quantization effects in metals or semiconductors have attracted considerable attention in the past decade [104—107]. Semiconductor nanoparticles may experience a transition in terms of electronic properties from those typical for a solid to that of a molecule, in which a complete electron delocalization has not yet occurred. These quantum-size effects arise when the Bohr radius of the first exciton (an interacting electron-hole pair) and the semiconductor becomes comparable with or larger than that of the particle the Bohr radius [94]... 

In conclusion of this section one may say that the linear optics of superlattices bear rich information on the dynamics of interfaces. Such investigations may give an idea of the nature of the interaction of bulk excitations (excitons) with interfaces which manifests itself in additional boundary conditions (ABC) and determines the value of the constant a. Finally, the study of dispersion laws for polaritons in superlattices with semiconductor layer thicknesses small in comparison with the Bohr radius of exciton permits one to follow variations in the properties of excitons. 

To calculate the effective transition dipole moment we need to know the wave-function of the exciton. In a quantum dot it depends on two interactions (i) the electron and hole confinement potential, which we shall assume to be infinite for r > Ri and zero for r < f i and (ii) the electron and hole Coulomb attraction. For these interactions we have to consider the following characteristic lengths f i - the radius of the quantum dot, and ag - the Bohr radius of an exciton in a macroscopic three-dimensional semiconductor. The problem of solving the two-particle Schrodinger equation for an arbitrary ratio of these lengths is quite difficult but the situation simplifies substantially in two important limiting cases. 

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