Fermi hole analysis

Fermi hole analysis is a very interesting tool for examining the degree of electron delocalization but has the drawback that it requires careful visual inspection and... 

Fermi hole analysis and MCI agree very well among each other in describing electron delocalization. 

Therefore, for all finite systems, the asymptotic behavior of v (r) arises from the Fermi hole charge distribution p r, r ) and is given exactly by the structure of WP(r). The above analysis and conclusions are borne out as shown in the example of the Helium ground-state discussed in Sect. 5.2.1. 

Pelster, A., Kleinert, H. (1996). Relations between Markov processes via local time and coordinate transformations. Phys. Rev. Lett. 78, 565-569. (arXiv cond-mat/9608120v2) Ponec, R., Cooper, D. L. (2007). Anatomy of bond formation. Bond length dependence of the extent of electron sharing in chemical bonds from the analysis of domain-averaged Fermi holes. FamrfqyDwcMss. 135, 31-42. 

Ponec R, Cooper DL (2007) Anatomy of bond formation. Bond length dependence of the extent of electron sharing in chemical bonds from the analysis of domain-averaged Fermi holes. Faraday Dis 135 31 2... 

Worth noting that such electronic density expression is of the first importance in characterizing the exchange or Fermi holes in chemical structures (Becke, 1986 Putz, 2008a), thus furnishing the backbone of the analytical chemical bonding analysis. 

Ponec, R. J. (1997). Electron pairing and chemical bonds. Chemical structure, valences and structural similarities fiom the analysis of the Fermi holes. Math. Chem. 21, 323-333. 

As tools for measuring electron delocalization, we opt for the visual inspection of the occupied orbitals and especially the so-called domain-averaged Fermi-hole (DAFH) analysis and MCI. Both of these require availability of atomic overlap matrices, where the overlap between two molecular orbitals is obtained in atom condensed form. This requires the definition of an atom in the molecule and in this study we chose the Ftirshfeld-I method. Finally, we introduce ring current maps in the ipso-centric approach. 

In addition to the numerical information provided by the values of various bond indices, the family of tools for the description of molecular structure was complemented some time ago by the new approach based on the analysis of the so-called domain averaged Fermi holes (DAFH). The most straightforward definition of these holes is via the restricted integration of the exchange part of the pair density [cf. eqn (2))... 

First, the carrier assignment is discussed using data of the hole-doped crystal. The dHvA frequencies corresponding to the a, Ji, y and 8 branches increase while those of the p branch decrease. This is due to the decrease in the Fermi energy by the hole-doping. In the dHvA specimen of the hole-doped crystal, chemical analysis indicates that the boron content is about 0.2% of the carbon sites, that is, 0.964 X 1020/cm3. Therefore it can be concluded that the a, Ji, y and 8 carriers are holes and the p carriers are electrons. 

Recombination is evidently controlled by trapping into defect states, consistent with the other recombination measurements. The recombination transitions through defects with two gap states are illustrated in Fig. 8.24, with electrons and holes captured into either of the two states. This type of recombination is analyzed by the Shockley-Read-Hall approach which distinguishes between shallow traps, for which the carrier is usually thermally excited back to the band edge, and deep traps, at which the carriers recombine. A demarcation energy, which is usually close to the quasi-Fermi energy, separates the two types of states. The occupancy of the shallow states is determined by the quasi-equilibrium and that of the deep states by the recombination processes. No attempt is made here at a comprehensive analysis of the photoconductivity, which rapidly becomes complicated. Instead some approximate solutions are derived which illustrate the processes. 

The change of units serves to emphasize the similarities between the analysis of semiconductors and electrolytic systems (see, e.g.. Chapter 5). The Fermi energy Ep is closely related to the electrochemical potential of electrons introduced in Section 5.2. Statistical mechanical arguments have been used to show that, under equilibrium conditions, the Fermi energy is equal to the electrochemical potential of electrons. At equilibrium, a single value of Fermi energy is sufficient to define the state of the system. Under nonequilibrium conditions, a separate Fermi energy can be defined for electrons and holes. 

The situation is quite different if minority carriers are involved. Then electrons and holes are not in equilibrium and their quasi-Fermi levels become different. In the case of an n-type semiconductor, f,p can be located above or below depending on the minority carrier process, i.e. on whether minority carriers are extracted from or injected into the semiconductor. However, quasi-Fermi levels have been qualitatively used in the theory of non-equilibrium processes in solid state devices, such as the excitation and recombination of electrons and holes (see Section 1.6), and also for the descriptions of charge transfer processes in p-n junctions (see Section 2.3). In this section a quantitative analysis of reactions at n- and p-type electrodes in terms of quasi-Fermi levels will be derived [19, 54]. 

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