Rigid band model

Fournier R and Salahub D R 1990 Chemisorption and magnetization A bond order-rigid band model Surf. Sol. 238 330-40... 

In this work, we present calculated SFE using the LKKR-CPA method for Al-Cu and Al-Mg which are of interest from the point of view of superplasticity. We use the SFE to validate the rigid band model which allows a deeper insight into the electronic structure and its implication on the nature of inter-atomic potentials. 

Figure 3 Compositional dependence of the stacking fault energy calculated from the rigid-band model (solid line) compared with the more accurate results from the LKKR-CPA calculation (dashed line) for the Al-Cu alloy system.

Parker [55] studied the IN properties of MEH-PPV sandwiched between various low-and high work-function materials. He proposed a model for such photodiodes, where the charge carriers are transported in a rigid band model. Electrons and holes can tunnel into or leave the polymer when the applied field tilts the polymer bands so that the tunnel barriers can be overcome. It must be noted that a rigid band model is only appropriate for very low intrinsic carrier concentrations in MEH-PPV. Capacitance-voltage measurements for these devices indicated an upper limit for the dark carrier concentration of 1014 cm"3. Further measurements of the built in fields of MEH-PPV sandwiched between metal electrodes are in agreement with the results found by Parker. Electro absorption measurements [56, 57] showed that various metals did not introduce interface states in the single-particle gap of the polymer that pins the Schottky contact. Of course this does not imply that the metal and the polymer do not interact [58, 59] but these interactions do not pin the Schottky barrier. 

One early and simple concept is the rigid-band model (Friedel 1958), wherein a fixed DOS is taken to represent an entire class of alloys (such as those composed of 3d transition metals). Individual alloys are distinguished solely by assigning to each a Fermi level, determined by the concentration of valence electrons. Unfortunately, this model is too much of an oversimplification, because, for example, the DOS is chosen empirically, and may not be clearly related to that for any of the constituent metals. 

Fig. 7.11 Density of states for amorphous Mg-Bi, rigid-band model. The gap may be...

Fig. 55. Debye temperature, d, and density of states at the Fermi level, N(Ep), for Y(Ni xCox)2B2C and Y(Ni xCux )2B2C as a function of the Co/Cu substitution level x. Symbols results derived from a relativistic band calculations in the atomic sphere approximation. Curves (in lower panel) rigid band model. After Ravindran...

The discovery by ultraviolet photoelectron spectroscopy that the rigid-band model is not even applicable to the bulk of alloys such as Ni-Cu and Pd-Ag. The d-electrons of copper or silver in such alloys experience a potential that is predominantly the same as in the metals before alloying. For instance, upon alloying the d-bands of copper and nickel remain discernible in the alloy and no common d-band is formed, as is supposed by the rigid-band theory. 

The work of both Bilz and Dempsey makes use of a rigid band model to determine the direction and extent of charge transfer. This model is inappropriate in the presence of a strongly perturbing influence such as interstitial C and N. This fact motivated the use of band structure techniques to once-and-for-all determine the direction and magnitude of charge transfer. Unfortunately, these investigations raised as many... 

It should be noted that the rigid band model and the tunnelling process discussed above are an idealization of the real device. It is unlikely that the barrier is exactly triangular as sketched in Fig. 5.2. The results presented in this book are aimed to give a further insight into the microscopic features of the metal-polymer interfaces and how these can be related to the macroscopic models such as the relations above. 

Possible explanations for this unusual behaviour have been offered by first-principles band structure calculations and synchrotron X-ray structural studies. The former revealed that, because of hybridisation between Ba and C orbitals, the rigid-band model is not appropriate for the description of the electronic properties and the calculated N(ev) for K3Ba3C60 and Rb3Ba3C60 are almost identical [69]. The structural analysis revealed positional disorder of the Ba2+ and K+ ions in the distorted tetrahedral sites of the bcc structure and the existence of short Ba-C and K-C contacts, consistent with strong hybridisation between the K, Ba and the C60 states [70]. It is important to notice that in K3Ba3C60 there is a perfect matching between the size of K+ and Ba2+ ions, while in both the Rb+ and Cs+ analogues, there is a considerable mismatch, which leads to fundamental structural... 

Figure 31 Rigid-band model of the energy levels in a diode (a) zero voltage (b) flat band (voltage Vn, applied) (c) large forward voltage V.

Assuming a fixed band structure (the rigid band model), a decrease in the density of states is predicted for an increase in the electron/atom ratio for a Fermi surface that contacts the zone boundary. It will be recalled that electrons are diffracted at a zone boundary into the next zone. This means that A vectors cannot terminate on a zone boundary because the associated energy value is forbidden, that is, the first BZ is a polyhedron whose faces satisfy the Laue condition for diffraction in reciprocal space. Actually, when a k vector terminates very near a BZ boundary the Fermi surface topology is perturbed by NFE effects. For k values just below a face on a zone boundary, the electron energy is lowered so that the Fermi sphere necks outwards towards the face. This happens in monovalent FCC copper, where the Fermi surface necks towards the L-point on the first BZ boundary (Fig. 4.3f ). For k values just above the zone boundary, the electron energy is increased and the Fermi surface necks down towards the face. 

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