Effective mass approximation

In the case of solids, the band dispersion describes a complicated dependence of energy on momentum, that usually cannot be described analytically. However, in the case of a semiconductor, the dispersion relations at the top of the valence band (TVB) and at the bottom of the conduction band (BCB), can often be described approximately as parabolic. Therefore, near the band edges, the delocalized electrons or holes, follow a quadratic equation of the form 

For an infinite potential outside the nanocrystals and zero potential inside, Efros and Efros [41], Brus [42-44] and Kayanuma [45, 46] proposed the following equation for the band gap, Er, of a quantum dot of radius R 

The EMA has been used to calculate the band gap for various semiconducting nanocrystals [47-53]. For larger sizes of nanocrystallites, the infinite potential (IP)- 

EMA gives a good description of the band gap variation vith size. However, it grossly overestimates the change, Eg, in the band gap for smaller nanocrystals. 

The resulting problem was solved variationally in the Hylleraas coordinate system. 

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The values of crystal size reported in this study varied between 6.6 and 10.6 nm. These values were based on the effective mass approximation, which overestimates the bandgap increase with decreasing crystal size, particularly for small sizes. 

This reported size is probably overestimated simple effective mass approximation used in its estimation. 

If the simple effective mass approximation does not hold, the expression 1 + mjmv is replaced by some other quantity, which also exceeds unity. 

Here 1 labels the electronic states of the SWCNT with the chiral index (p,0), which are described by a simple two-band k p model based on an effective mass approximation [4], p being equal to 3M + v with integer M and v = 0( 1) for metallic (semiconducting) SWCNTs. The energy bands in Eq.(3) are given by... 

Exciton dispersion As the 0-0 exciton dispersion in anthracene is somewhat complex,101 we assumed a bidimensional parabolic dispersion (effective-mass approximation). This assumption is justified by the absence of interplane coupling for k perpendicular to plane (001) planes2 9—the case of b-polarized exciton under study—and it is aimed to give back the observed stepwise thresholds bound to the bottom of the band. The model most likely fails for the exciton band as a whole, but anyhow, we could not avoid it without lengthening computation times excessively. 

It should be noted that spectroscopic data obtained over a wide range of a values show that Vi is not quite linear when plotted against 1/a, and the observed curvature is greater than that due to the coulombic term shown explicitly in Eq. (9). Such curvature is due to nonparabolic band contours and to a breakdown of the effective mass approximation. As a result, the filnig values obtained with Eqs. (7) and (9), and to a lesser extent the Epoi value, will depend on the range of a values used in the fits. 

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. 

Stable, cubic phase, PbS nanoparticles were prepared in a polymeric matrix by exchanging Pb + ions in an ethylene - 15% methacrylic acid copolymer followed by reaction with H2S [91]. The size of the PbS nanoparticles was dependent on the initial concentration of Pb + ions with diameters ranging from 13 to 125 A. The smallest particles (13 A) are reported to be molecular in nature and exhibit discrete absorption bands in their optical spectra. Two theoretical models, which take into account the effect of nonparabolicity, were proposed in order to explain the observed size-dependent optical shifts for PbS nanocrystallites. The authors reported that the effective mass approximation fails for PbS nanocrystallites. 

A lot of theoretical work has been done in order to explain the size dependent properties of semiconducting nanocrystals. These methods are primarily based on the effective mass approximation, pseudopotential approaches or the tight binding scheme. Each of these methods has certain advantages and disadvantages. We shall explore these methods in some detail in Section 11.5. 

The correlation between the energy of the band gap and the size of the nanoparticle can be described using one of two theoretical models [74, 75, 77]. The first model is a modification of the effective mass approximation, in which the Cou-lombic interaction between the electron and hole are taken into consideration. In this approach, the valence and conduction bands are assumed to be parabolic near the band gap, yielding Eq. (5) for the difference in energy of the first excited state, relative to the bulk [74] ... 

The values me and mh are the effective masses of the electron and hole respectively. The second term describes the Coulombic interaction of the electron and the hole, where e is the universal charge and e is the dielectric constant of the material in question. Good correlation between theory and experiment has been obtained within this model for larger nanoclusters [78]. In the case of smaller particles, especially those smaller than 2 nm, the lowest exdted states are located in a region of the energy band that is no longer parabolic and, as a result, the effective mass approximation breaks down. For these partides a more molecular approach re-... 

The earliest and simplest treatment of the electronic states of a QD is based on the effective mass approximation (EM A) the simple EM A treatment can be improved by incorporating the k p approach, which has commonly been nsed to calcnlate the electronic stmctnre of bnlk semicondnctor and QW stmctnres. 

In the effective mass approximation we assume that the expansion (4.121) is valid in the energy range (near the conduction band edge) for which the integrand in (4.127) is appreciable. In this case we can use for E) the free particle expression (cf. Eq. (4.59))... 

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