Electrons effective mass theory

The Coulomb interaction between the electron and the donor core is, of course, present in an amorphous semiconductor and binds an electron in much the same way, so the shallow donor state is preserved. The effective mass theory for dopants cannot be applied directly to amorphous semiconductors, because it is formulated in terms of the momentum-space wavefunctions of the crystal. It is not immediately obvious that the effective mass has any meaning in an amorphous... 

The electronic structure is reformulated in terms of free electrons and a d resonance in order to relate the band width W, to the resonance width T, and is then reformulated again in terms of iransilion-metal pseudopotential theory, in which the hybridization between the frce-electron states and the d state is treated in perturbation theory, The pseudopotential theory provides both a definition of the d-state radius and a derivation of all interatomic matrix elements and the frce-electron effective mass in terms of it. Thus it provides all of the parameters for the L.CAO theory, as well as a means of direct calcidation of many properties, as was possible in the simple metals. ... 

Banin U., Lee J. C., GnzeUan A. A., Kadavanich A. V., Alivisatos A. P., Jaskolski W., Bryant G., Efros A. L. and Rosen M. (1998), Size-dependent electronic level structure of InAs nanocrystal quantum dots test of multiband effective mass theory , J. Chem. Phys. 109, 2306-2309. 

Simple approach based on the effective mass theory has been developed and successfully applied to simulate electronic properties of monocrystalline and grained nanocrystalline films accounting for the confinement effect and interactions between the grains. Quantum confinement was found to influence band gap values only for the films with the thickness less than 5 nm. The highest gap varied from 0.63 to 0.91 eV depending on the film thickness as well as on the lateral size of the grains. Inclusion of the grains inside the film induces a eonsiderable increase of the gap as compared to the monocrystalline film of the same effective thickness. 

The electronic properties of monocrystalline and grained nanocrystalline CrSi2 films were estimated within the Effective Mass Theory. Inclusion of the grains inside the film increases the energy gap up to 60% compared to the monocrystalline film of the same effective thickness. 

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. 

During this lecture I wish to discuss how has been calculated in the case of liquid Argon (Lekner, 1967 Plenkiewitz et al., 1986). When the energy of states of non-zero wave vector is calculated the effective mass is obtained in the same way as for electrons in a crystalline solid (Kittel, 1963). The effect of density fluctuations can be easily estimated in the framework of the effective mass theory. 

Historically, solid state theory has been dominated by physicists. One consequence of this is that the conventional methods which resulted have tended to emphasize physical rather than chemical properties, such as band structures, electron effective masses, and densities of states, rather than bond orders, geometries, and electron distributions. Another consequence of the way these methods developed is that the... 

As mentioned above, the results discussed below are obtained using Ab initio methods. Other methods used to study QDs are effective mass theory (EMT) and the pseudopotential techniques. EMT uses a particle-in-a-box model where the electron and hole masses are given by their bulk values. EMT is an intuitive description that explains general trends seen in experiments. The atomistic pseudopotential technique can be applied to large systems, but requires careful parameterization for each material. Ab initio approaches use minimal parameterization and are applicable to most materials. This makes them particularly useful for studying dopants, defects, ligands, core/shell systems and QD synthesis. The Hartree-Fock (HE) method and density functional theory (DFT) have been around for many decades, while time domain (TD) DFT and non-adiabatic molecular dynamics (NAMD) are more recent areas of research. Currently, ab initio TDDFT/NAMD is the only... 

Figure 12-5. Kcprcscmauun of Uie calculated injcciiou curretu on a 111 j vs scale. Tlic dashed line indicates tile slopes predicted by Fowler Nordheiin tunneling theory lor A=0.8eV assuming that the effective mass equals the free electron mass.

The Schottky-Mott theory predicts a current / = (4 7t e m kB2/h3) T2 exp (—e A/kB 7) exp (e n V/kB T)— 1], where e is the electronic charge, m is the effective mass of the carrier, kB is Boltzmann s constant, T is the absolute temperature, n is a filling factor, A is the Schottky barrier height (see Fig. 1), and V is the applied voltage [31]. In Schottky-Mott theory, A should be the difference between the Fermi level of the metal and the conduction band minimum (for an n-type semiconductor-to-metal interface) or the valence band maximum (for a p-type semiconductor-metal interface) [32, 33]. Certain experimentally observed variations of A were for decades ascribed to pinning of states, but can now be attributed to local inhomogeneities of the interface, so the Schottky-Mott theory is secure. The opposite of a Schottky barrier is an ohmic contact, where there is only an added electrical resistance at the junction, typically between two metals. 

One further effect of the formation of bands of electron energy in solids is that the effective mass of electrons is dependent on the shape of the E-k curve. If this is the parabolic shape of the classical free electron theory, the effective mass is the same as the mass of the free electron in space, but as this departs from the parabolic shape the effective mass varies, depending on the curvature of the E-k curve. From the definition of E in terms of k, it follows that the mass is related to the second derivative of E with respect to k thus... 

The failure is not limited to metal-ammonia solutions nor to the linear Thomas-Fermi theory (19). The metals physicist has known for 30 years that the theory of electron interactions is unsatisfactory. E. Wigner showed in 1934 that a dilute electron gas (in the presence of a uniform positive charge density) would condense into an electron crystal wherein the electrons occupy the fixed positions of a lattice. Weaker correlations doubtless exist in the present case and have not been properly treated as yet. Studies on metal-ammonia solutions may help resolve this problem. But one or another form of this problem—the inadequate understanding of electron correlations—precludes any conclusive theoretical treatment of the conductivity in terms of, say, effective mass at present. The effective mass may be introduced to account for errors in the density of states—not in the electron correlations. 

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