Electron effective mass approximation

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... 

Interestingly enough, it has been possible to write the free-electron effective mass in terms of the same r/-state radius that determined the d-band width and determined hybridization with the free electrons. It was noted in the discussion of the Atomic Sphere Approximation that Andersen defined an effective mass for d electrons in terms of the r/-band width, = 12.5fiV(m, ro) This can be combined with the expression for in terms of (Eq. 20-9), to write m /m = (1 + 2.9lm/m ). In the Atomic Sphere Approximation Wp, and arc regarded as independent quantities, but both the and values given by Andersen and Jcpsen (1977) are rather close to the effective mass m., obtained from m,/m = (I + 2.91j /ni,j) . ... 

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. 

Quantum films represent a beautiful physical example of the particle in the box problem. The energy levels of the conduction band electrons in the electron well (MQW) can be easily calculated using the envelope function or the effective mass approximation [6]. The electron wave function is then [29-31]... 

Many models have been presented to explain quantitatively the dependence of exciton energy on the cluster size [7, 11, 21-30]. This problem was first treated by Efros et al. [7], who considered a simple particle in a box model. This model assumes that the energy band is parabolic in shape, equivalent to the so-called effective mass approximation. The shift in absorption threshold, AE, is dependent upon the value of the cluster radius, R, Bohr radius of the electron, ae ( = h2v.jmce2), and Bohr radius of the hole, ah (= h2e/mhe2). When (1) R ah and R ae, and (2) ah R ac,... 

Small clusters frequently have structures that are different from that of the bulk and consequently have different effective masses for electrons and holes. It is not possible to handle such structural deviation within the effective mass approximation model. 

If the bands are parabolic (i.e. E k2) and we are within the effective mass approximation (for which it is assumed that all electron-electron and electron-nuclear interactions can be absorbed into an effective electron mass), explicit expressions for e2(co) can be obtained. These expressions will be most accurate near the so-called critical points, which are points in the energy spectrum at which a new transition has its onset or disappearance, or at which there is a change in the type of transition observed. The nature of a critical point is illustrated in Fig. 8, which shows a valence band and a wider conduction band. For this very simple system, there are four critical points, labelled A-D in the figure. The valence band (VB) and conduction band (CB) are characterised, within the effective-mass approximation, by effective masses m and m, where... 

Electron and hole states have been calculated for Si nanowires and quantum dots within the effective mass approximation. In the calculation of the electron states, six anisotropic valleys of bulk Si have been taken into account. It is found that the states depend on the crystallographic direction and on the size of the wires and the dots. These results have been used to calculate electron-hole recombination lifetimes. The magnitudes of the lifetimes are very s ensitive t o t he g eometrical a nd s tructural p arameters o f t he w ires and t he d ots. 11 i s concluded that non-uniformity in the crystallographic direction of Si nanowires and quantum dots causes itself dispersion in the values of the photoluminescence lifetime. 

A simple model of a nano-dimensional structure in the form of a neutral spherical SNc of radius a and permittivity si, embedded in a medium with permittivity ej, has been discussed elsewhere. An electron e and hole h with elfective masses and m i were assmned to travel within this SNc (we use r and rh to denote the distances of the electron and the hole, respectively, from the center of the SNc). We assume that the two permittivites are such that E2 ei, and that the electron and hole bands are parabolic in shape. In this model, and subject to these approximations and the effective mass approximation, the exciton Hamiltonian takes the... 

Very delocalized excitons are treated in the Waimier effective mass approximation. The hole and the electron interact by the coulombic attraction, modified by an appropriate dielectric constant k. The effects of the periodic crystalline potential show up only in a reduced mass 11 (which may be anisotropic) containing the effective masses of the hole and the electron. Excitonic states are then hydrogen-like, with binding energies E (with respect to the conduction-band... 

A very important step toward a better understanding of the electronic properties of layered semiconductor particles is presented in the same article. In the framework of the effective mass approximation, the authors modeled the optical transition of CdSe crystallites with small inclusions of... 

Some photobleaching experiments have been carried out with QDQWs [58]. They reveal that the photobleaching spectrally follows the newly evolving Is-Is electronic transition of the composite particles, thus pointing also in the direction that the electronic structure of QDQWs is not simply a superposition of the electronic properties of the segments of the particles. This is also concluded from theoretical modeling of the particles in the framework of the effective mass approximation [57]. This calculation fits the Is-Is optical transition of the QDQWs of various compositions very well. A further description of the model goes beyond the scope of this chapter. 

It is assumed that a source of charge carriers is the local electronic states in the nanotube-electrode interface layer, the electrons from which emerged to the conduction band of the nanotube crystal due to the PhAT from these centers. If electrons released from these centers dominate the current through the crystal, I will be proportional to the electron released rate IT and the density of the centers N, i.e. I x NW. For the calculation of W with participation of phonons we operate with the PhAT constructed in the effective mass approximation. The tunneling rate W(E,T) has been derived as [13] ... 

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