Band theory effective 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... 

Optical properties of metal nanoparticles embedded in dielectric media can be derived from the electrodynamic calculations within solid state theory. A simple model of electrons in metals, based on the gas kinetic theory, was presented by Drude in 1900 [9]. It assumes independent and free electrons with a common relaxation time. The theory was further corrected by Sommerfeld [10], who incorporated corrections originating from the Pauli exclusion principle (Fermi-Dirac velocity distribution). This so-called free-electron model was later modified to include minor corrections from the band structure of matter (effective mass) and termed quasi-free-electron model. Within this simple model electrons in metals are described as... 

We do this by using the k p method, (called k-dol-p), which is based upon the perturbation theory of Eq. (1-14). In this method, energy is calculated near a band maximum or minimum by considering the wave number (measured from the extremum) as a perturbation. (The method is described in many solid state texts, such as Kittel, 1963, p. 186, or Harrison, 1970, p. 140.) The method was used for a study of effective masses by Cardona (1963, 1965). It was also usetl in the more extensive study by Lawaetz (1971) referred to in the discussion of heavy-hole bands. We shall discuss here only the conduction band and the light-hole band where the effects of interaction are great. 

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. ... 

For analysis of the transition metals themselves, the use of free-electron bands and LCAO d states is preferable. The analysis based upon transition-metal pseudopotential theory has shown that the interatomic matrix elements between d states, the hybridization between the free-electron and d bands, and the resulting effective mass for the free-electron bands can all be written in terms of the d-state radius r, and values for have been listed in the Solid State Table. 

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-... 

Theory of excitons often uses the so-called effective mass approximation, assuming the energy dispersion as in eqn (1.6), where only the terms displayed on the r.h.s. are taken into account. In this approximation the effective mass also determines the width of the exciton energy band, which for isotropic bands is of order... 

Since shallow-level impmities have energy eigenvalues very near those of the perfect crystal, they can be described using a perturbative approach first developed in the 1950s and known as ejfective mass theory (EMT). The idea is to approximate the band nearest to the shallow level by a parabola, the curvature of which is characterized by an effective mass parameter m. ... 

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. 

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