Hole distribution function

Essentially, it is determined by the combined density-of-states of the conduction and valence bands and by the difference of the electron and hole distribution functions. In fact, the effect of the band structure is two-fold. On one hand a large combined density-of-states is beneficial for a large optical gain, and on the other hand large densities-of-state correspond to large carrier densities in order to achieve inversion. 

To obtain the spin-Hall current, it is necessary to analyze the hole distribution function. Within the linear response regime, we find that the linear electric field part of the helicity-basis distribution function, j(p), can be written as a sum of two terms,... 

Fig. 10.9. (a) Calculated optical absorption spectrum of (rans-polyacetylene from a DFT-GWA-BSE calculation. The solid and dashed curves represent the exciton and quasi-particle spectra, respectively, (b) The electron-hole distribution function. Reprinted with permission from M. Rohlfing and S. G. Louie, Phys. Rev. Lett., 82, 1959, 1999. Cop5rright 1999 by the American Physical Society. 

Here u fl" and E " are the periodic part of the Bloch function, energy and Fermi-Dirac distribution functions for the n-th carrier spin subband. In the case of cubic symmetry, the susceptibility tensor is isotropic, Xcj) = Xc ij- It has been checked within the 4 x 4 Luttinger model that the values of 7c, determined from eqs (13) and (12), which do not involve explicitly u and from eqs (14) and (15) in the limit q - 0, are identical (Ferrand et al. 2001). Such a comparison demonstrates that almost 30% of the contribution to 7c originates from interband polarization, i.e. from virtual transitions between heavy and light hole subbands. 

In the related field of foam stability, De Vries (D3) has performed some very interesting studies. He suggests that the formation of a hole in the separating film and also the first expansion of this hole requires an increase of free energy. This activation energy, which must be supplied in order to make expansion possible, is proportional to the square of the film thickness and to the interfacial tension. The chance that this activation energy is indeed supplied is then described by a Boltzmann distribution function. 

It is worth remembering that we are still working with the one-electron picture, and that we have applied the Boltzmann relation in order to approximate Fermi and quasi-Fermi distribution functions, assuming the quasi-free electron and hole densities of states in the bands. 

The luminescence efficiency is given by the fraction of electron-hole pairs which are created farther than R from the nearest defect (Street et al. 1978). The distribution of distances is the nearest neighbor distribution function, G R), for randomly dispersed defects, which is (Williams 1968)... 

Fig. 10 under equilibrium i. e., the Fermi levels are the same in both phases (compare with Fig. 9). Because of a gap in the distribution function of electron levels where the Fermi level is normally found, exchange of electrons cannot occur in this region. Exchange occurs only at energy levels in two separated energy regions, the conductance band and the valence band, where the number of exchangeable levels is very small compared with the region where exchange takes place at a metal. Therefore the total exchange rate must be very much smaller at a semiconductor. The electron exchange in the valence band can best be discussed in terms of minority carriers (holes).

To quantify the hole model, it is necessary to calculate a distribution function for the hole sizes. This is a plot of the number of holes per unit volume as a function of their size. As a first step toward this calculation, one can consider a particular hole in a liquid electrolyte and ask What are the quantities (or variables) needed to describe this hole This problem can be resolved by means of a formulation first published by Fiirthin 1941. 

Since the desired distribution function only concerns the radii (or sizes) of holes, it is sufficient to have the probability that the hole radius is between r and r + dr irrespective of the location and the translational and breathing momentum of the hole. This probability Pr dr of the hole s radius being between r and r + dr is obtained from Eq. (5.16) by integrating over all possible values of the location, and of the translational and breathing momentum of the hole, i.e.,... 

This is the distribution function which was said to be the goal at the beginning of Eq. (5.17). From it, the average hole volume and radius will shortly be seen to be obtainable (see Fig. 5.22). 

In Ftirth s theory of cavities in liquids, there is a distribution function for the probability of the hole size. It is... 

Using the distribution function, make a plot of probability that a hole has a radius in molten sodium chloride at 1170 K. The surface tension of molten sodium... 

In the Fiirth hole model for molten salts, the primary attraction is that it allows a rationalization of the empirical expression = 3.741 r p. In this model, fluctuations of the structure allow openings (holes) to occur and to exist for a short time. The mean hole size turns out to be about the size of ions in the molten salt. For the distribution function of the theory (the probability of having a hole of any size), calculate the probability of finding a hole two times the average (thereby allowing paired-vacancy diffusion), compared with that of finding the most probable hole size. 

When r > 0 K, the population of the electron energy levels is described by the Fermi-Dirac distribution function (See Section 4.3.3, Eq. 22). At T > 0 K, electrons from the valence band can be thermally excited into the conduction band. As a result, the bottom of the CB becomes partly populated and the top of the VB partly depopulated [Figure 7(b)]. An empty electron level at the top of the valence band is called a (valence-band) hole. The concentration of holes, p, and of electrons, n, can be expressed as a function of the electrochemical potential with Eq. 22. We denote the density of electron levels within IcbT from the top of the VB and the bottom of the CB as the effective density of valence band levels, Avb, and conduction band levels, Nqb, respectively. The electron occupancy of the electron levels at the bottom of the CB is... 

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