Hole number density

Bohlen, J., Wolff, J., Kirchhelm, R. (1999) Determination of free-volume and hole number density in polycarbonates by positron lifetime spectroscopy . Marcromolecules, 32,3766. 

For a binary gas mixture of electrons and holes, we define n and p as the electron and hole number densities, respectively, and also use the same letters as subscripts to represent the electron and hole components. Then we can write the charge Equation 5 for holes of charge +q, and electrons of charge -q, respectively ... 

Thus, an orbital description of the motion of an electron contains the same information conveyed by the holes made by darts in a dartboard. After the board has been used in many games, the distribution of holes shows how successful earlier players had been in their scoring. There are many holes near the bullseye and, moving away from it, there is a regular decrease in the number of holes per square centimeter of dartboard. At any given distance from the bullseye, the density of dart holes (number per square centimeter) is a measure of the probability that the next throw will (and there. 

The difference between a photoconductive detector and a photodiode detector lies in the presence of a thin p-doped layer at the surface of the detector element, above the bulk n-type semiconductor. Holes accumulate in the p-layer, and electrons in the n-type bulk, so between the two there is a region with a reduced number density of carriers, known as the depletion layer. The important effect of this is that electron-hole pairs, generated by photon absorption within this depletion layer, are subjected to an internal electric field (without the application of an external bias voltage) and are automatically swept to the p and n regions, and... 

Now think of the n —>p hole current. When the holes from the n-type of medium reach the junction [see Fig. 7.21(c)], they do not see any barrier due to an electrical potential difference, so they simply tumble over the potential drop. Hence, the n —> p hole current density at equilibrium is controlled only by diffusion and is simply proportional19 to the number of holes, nhn, in the rz-type of material ... 

But the holes crossing from the n- to p-type of material still have no barrier to climb. Hence, then— p hole current density still depends only on the number of holes in the 7 i-type of material and... 

Consider a polymer is quenched from liquid to glass where the sample is annealed. During isothermal annealing, the number of holes is close to a conserved quantity. The local excess of number density of the quenched glass relaxes by spreading slowly over the entire region, and is governed by... 

The flux of molecules to a surface, JN, determines either the rate at which it would be covered if each molecule stuck, or the rate of effusion through a hole in the surface. In studying the exposure, JNdt, of a surface to a gas, surface scientists find it useful to use the product of pressure and time as a measure of the exposure since this product is proportional to the number flux, JN, times the time JNt = (1/4)Cut = (u/4kT)pt, where C is the number density of molecules, u their average speed, k the Boltzmann constant and T the thermodynamic temperature. The unit langmuir (symbol L) corresponds to the exposure of a surface to a gas at 10" 6 torr for 1 second. 

Alternative ways of determining the free volume fraction without using I3 have also been proposed by Dlubek et al [28], as well as, Brandzuch et al [29], Dlubek et al used the coefficient of thermal expansion of the amorphous regions and hole volume determined from positron data to determine the number density of the free volume holes. Brandzuch et. al. used the coefficient of thermal expansion just above and just below the Tg to estimate the fractional free volumes. This model is based on the assumption that the expansion of the holes of the free volume, as seen by positrons, reflects the expansion of the total volume of the material. 

Excitation of the electron from the VB leaves behind a vacancy or "hole . Electrons in the VB can move to fill these holes leaving, in turn, further holes conceptually, it is easier to consider the holes moving in a sea of immobile electrons and we may develop expressions for the number density of these holes in close analogy to the formulae above for the electrons. Thus, the number density of holes in the VB at energy E, p(E ), is given by... 

The total number density of electrons excited, ni can be obtained by integrating over the energy range E = Ec to the top of the CB and, in a similar way, the total density of holes, p, is obtained by integrating eqn. (3) from the top of the VB to Ev. Clearly, these integrals cannot be performed unless the functional forms of Nv and Nc are available in practice, this is rarely so, but it is often found that the top of the valence band and the bottom of the conduction band have simple parabolic shapes of the form... 

The carrier number density of electi-on-hole pairs, Nc, is determined from the following conservation law given by... 

Although the lattice heat capacity in a metal is much larger than its electronic contribution, the Fermi velocity of electrons (typically 106 m/s) is much larger than the speed of sound (about 103 m/s). Due to the higher energy carrier speed, the electronic contribution to the thermal conductivity turns out to be more dominant than the lattice contribution. For a semiconductor, however, the velocity is not the Fermi velocity but equal to the thermal velocity of the electrons or holes in the conduction or valence bands, respectively. This can be approximated as v /3kBT/m, where m is the effective electron mass in the conduction band or hole mass in the valence band. This is on the order of 105 m/s at room temperature. In addition, the number density of conduction band electrons in a semiconductor is much less than... 

To obtain the Vq value for the parent molecular beam, the speed profile for the parent molecular beam obtained by the chopper wheel or the laser hole-burning method is fitted to an assumed number density distribution of the form f v) r exp[ —(r — Lo)V(Ai ) ], where Av is a measure of the width of the speed profile. 

In analogy to previous results [107, 114—120], all PS films on such substrates turned out to be metastable and ruptured upon annealing at elevated temperatures. The number density of holes increased rapidly with decreasing film thickness [107]. We concentrated on the shape of these holes and its evolution with time at temperatures above about 103°C. Complementary experiments indicated, however, that hole formation was also possible at lower temperatures [162]. As discussed previously [4, 17], hole formation may also reflect the relaxation of internal tensions induced during sample preparation and caused by confining the polymers to films thinner than the size of the unperturbed coil. 

For the two-carrier system, the situation is considerably different. In this case, referring to Fig. 3b, thermal energy is transported by both holes and electrons and the Ettingshausen effect is mainly due to generation of hole-electron pairs at one side of the sample and their recombination on the other side. In addition, if the number densities and the mobilities of the holes and electrons are equal, the Hall effect and thus the opposing Hall field vanishes. In this case, the Ettingshausen effect is expected to be quite large. 

An alternative approach is to make use of the o-Ps intensity h. This parameter has been proposed [Wang et al., 1990] to be linearly correlated to the number density of free-volume holes, so that the free-volume fraction can be written only in terms of parameters derived from PALS ... 

TABLE 10.1 Number Density of Holes N, Occupied Volume and Aspect... 

TABLE 10.2 Molecular Weights, Glass TVansition Temperature Tg, Characteristic Reducing Parameters V and T, Exponent k, Number Density of Holes N, and Experimental and Theoretical (V ) Occupied Volume for the Perfluoropolyethers Investigated... 

The crosses in Figure 11.9 show the free volume for PC calculated from Eq. (11-5) and the determined o-Ps hole size (u/,). The specific hole number A was used as a fitting parameter. A constant value of A = 0.67 x 10 was sufficient for the temperature range between 300 and 570 K to get agreement between the free volumes estimated from = A (w ) and Vf = hV. The volume- elated hole density A = A /y varies with T. Therefore, the hole volume ( > cannot be scaled to the hole fraction h to determine Aft [Higuchi et al., 1995 Schmidt and Maurer, 2000b Kilbum et al., 2005]. 

The Specific hole number at Tg, of conventional polymers shows a linear increase with Tg (Figure 11.12b) from 0.2 to 0.8 in units of 10 g (for polymers of density 1, this corresponds to the same numbers in units of nm ). The fiuoropoly-mers show an almost constant hole density 0.2 x 10 corresponding to... 

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