Effective-Mass Theory and its Use

A Question of Overlap. Previous experience with low-temperature superconductors, and with the BCS theory used to analyze them, has made it almost a matter of dogma that large effective mass m (and consequently, large density-of-states.) are conducive to high-temperature superconductivity. Nevertheless, here the facts dictate otherwise (8,9). The concensus for the htgh-Tc materials —... 

This chapter summarizes the main theoretical approaches to model the porous silicon electronic band structure, comparing effective mass theory, semiempirical, and first-principles methods. In order to model its complex porous morphology, supercell, nanowire, and nanocrystal approaches are widely used. In particular, calculations of strain, doping, and surface chemistry effects on the band structure are discussed. Finally, the combined use of ab initio and tight-binding approaches to predict the band structure and properties of electronic devices based on porous silicon is put forward. 

Effective Mass Theory (EMT). Historieally, the eoneept of effeetive mass was very successful to understand the eleetronic behavior in semieonduetors and to design electronic devices. Nowadays, in the study of nanostruetures, the EMT is still useful to describe qualitatively the quantum confinement effects, but it overestimates the quantitative results. For instance, when pSi is modeled by SiNW, the bandgap (E g) as a funetion of its diameter d) follows EQ d) = Eq o6) + C( ldf, where C and a are positive eonstants, and EMT gives the upper limit of a = 2. This overestimation can be related to EMT ignoring the electronic density fluctuations before the quantum confinement. 

It should, however, be mentioned that Mulliken s study of the BO system has been followed over the years by many others, An extensive study by Jenkins and McKellar (1932) should be mentioned explicitly. This study involved the long wavelength band of BO. The same method as that used by both Jevons and Mulliken to produce the BO was used in this work. The new (present day) quantum mechanics was used in the theoretical interpretation. Both the vibrational and the rotational isotope effects were observed and agree with theory. One motivation for this work was to determine how well the isotopic ratio of the square roots of the two relevant isotopic masses (10B and nB) agrees with the ratio obtained from Aston s mass spectrometric measurements and hence how well isotopic mass ratios determined from band spectra compare with those obtained using Aston s mass spectrograph. 

The expression for the enhancement factor E, eq. (35), has first been derived by van Krevelen and Hof-tijzer in 1948. These authors used Pick s law for the description of the mass transfer process and approximated the concentration profile of component B by a constant Xb, over the entire reaction zone. It seems worthwhile to investigate whether the same equation can be applied in case the Maxwell-Stefan theory is used to describe the mass transfer process. To evaluate the Hatta number, again an effective mass transfer coefficient given by eq. (34), is required. The... 

Figures 7(a)-(c) show a comparison between the numerically computed absorption flux and the absorption flux obtained from expression (31), using eqs (24), (30) and (34)-(37). From these figures it can be concluded that for both equal and different binary mass transfer coefficients absorption without reaction can be described well with eq. (24), whereas absorption with instantaneous reaction can be described well with eq. (30). If the Maxwell-Stefan theory is used to describe the mass transfer process, the enhancement factor obeys the same expression as the one obtained on the basis of Fick s law [eq. (35)]. Finally, from Figs 7(b) and 7(c) it appears that the use of an effective mass transfer coefficient m the Hatta number again produces satisfactory results.

A quantity, /x, used in collision theory for the collision of two molecules having masses mA and OTb It is equal to mAmB/ipiA + OTb). See Hooke s Law Spring Kinetic Isotope Effects... 

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