Number of Conduction Electrons

This section follows directly from Section 4.8. The number of electrons in the conduction band, n is 

A( ( ) dE is the density of states in the conduction band and represents the number of energy levels over which 

E - E kT. This is often the case since at room temperature kT 0.025 eV and E E[is usually 5 eV. We can now omit the -i-l in Eq. 30.7. [We are in effect replacing Fermi-Dirac statistics by Boltzmann statistics.] 

The excited electrons occupy states near the bottom of the conduction band. Under these conditions they behave as free particles for which the state distribution function is known. 

The upper limit of the integration in Eq. 30.6 is taken as since the probability of occupancy of a state by an electron rapidly approaches zero as the energy increases through the band. 

---

For local deviations from random atomic distribution electrical resistivity is affected just by the diffuse scattering of conduction electrons LRO in addition will contribute to resistivity by superlattice Bragg scattering, thus changing the effective number of conduction electrons. When measuring resistivity at a low and constant temperature no phonon scattering need be considered ar a rather simple formula results ... 

If we say that, with a given applied field, in unit time this excess consists of n electrons, the current density will be Ne, since we are dealing with unit area. In Fig. 16 let us suppose that the excess flow of electrons is in the downward direction we can then, to show the character of the flow, make the following construction. Parallel to the plane AB, consider a plane CD, also of unit area and let the distance between CD and AB be chosen such that the total number of conduction electrons in the volume between CD and AB at any moment is n. 

In accordance with Ohm s law, if we were to double the intensity X of the electric field, the current would be doubled that is to say, the plane CD would have to be placed at twice the distance from AB. If the number of conduction electrons per unit volume is p, and the distance between the planes CD and AB is denoted by v, we have n = pv, since we are discussing the unit area. Hence the net resultant charge transported in unit time across AB, that is, the current density, is given by... 

Referring to results obtained in study [16], we can assume that the conductivity of crystals in sintered ZnO film increases due to increase in number of conductivity electrons in the surface layer. Taking the sample mobility of electrons fd as 10 cm -s -V the temperature dependence being T (the data borrowed from [21]) one estimate the value A[e] from the following expression Acr = A[e, where Acr is the con-... 

The magnitude of the drift velocity vector v of conduction electrons can be calculated rather easily. It is surprisingly low. If we denote the number of conduction electrons per cubic meter n and the electronic charge as e, then... 

The explanation of this peak is as follows. Suppose that the number of conduction electrons is small, so that the Coulomb field is not screened out and that a hole in the X-ray level creates an exciton level below the bottom of the conduction band. The levels are shown in Fig. 2.13. Then an exciton absorption line should be possible. But the sudden change in field will produce excitations of electrons at the Fermi level, so that the exciton line is broadened as shown in Fig. 2.14(a). Also, we do not expect a sharp increase in absorption when the electron jumps to the Fermi level, leaving the exciton level A in Fig. 2.13 unoccupied, because of the very large Auger broadening due to transitions from the Fermi level into this unoccupied state. 

The variation in the measured electron mobilities from sample to sample in sintered materials (also observed by Hahn, ref. 24), may be due to any of several effects. The most probable reason for this variation in the well-sintered samples studied is a difference in history the individual samples are obtained with different numbers of conduction electrons per cm. frozen in in the necks. That is, the different history has allowed different amounts of oxygen to be adsorbed on the surface. Thus the concentration of electrons in the grain, as measured by the Hall coefficient, will have little relation to the concentration of electrons in the neck, as measured by the conductivity, and the mobility, obtained from the product of the Hall coefficient and the conductivity, will be neither the true mobility nor constant from sample to sample. The different samples may also end up with varying geometry of their necks, according to their previous treatment. 

For conductors the valence band is completely filled and the conduction band is only partially filled (Fig. 2a). This means that there are many carriers. A semiconductor possesses a completely filled valence band and completely empty conduction band at 0° K (Fig. 2 b) but at higher temperatures the number of conduction electrons is controlled by the distribution of electrons between the valence and conduction band. This in turn is controlled by the width of the energy gap and the density of states curve, i. e. the number of allowed states for an energy lying within the range E + 6E (Fig. 2 c). 

With this understanding, it is clear that in a given conduction 0 covalent transformation, a decrease or increase in the number of conduction electrons is an essential feature that should be observable in the transport properties. Assuming that the band structure of TiNi consists of a single positive band, a decrease in the number of conduction (free) electrons in the course of Ms —> As is equivalent to an increase in the number of hole carriers as seen in (c). Consequently, the positive Hall coefficient should decrease and is so observed in (b). Because holes contribute to Pauli paramagnetic susceptibility in precisely the same manner [42] as electrons, the paramagnetic susceptibility, %, is expected to rise and is so observed in (d). An increase in the hole carrier, Nh, would result in an increase in the conductivity (lowering in the resistivity) as... 

Here, Nc is the number of conductivity electrons containing in unit of volume, cop = (Nce2/m)1/2 the plasma frequency of metal, and E — Eext exp —icot. The real part of coefficient at E in the formula (4) is the part of dielectric permeability s c caused by conductivity electrons (s c — —co2/co2 + r2). Total dielectric permeability s is equal to... 

Metals are characterized by a plasmon frequency Ep proportional to the square-root of the number of conduction electrons per A3. Metallic lithium (162) shows E-p—7.5 and 2.Ep = 14.8 eV above / (Lils) =54.8 eV and sodium (162, 163) five almost equidistant plasmon signals with consecutive distances 5.8 to 5.9 eV... 

In view of the merely approximate validity of equation (7), from which is calculated, and the experimental error in Eq, equation (6) can only give the right order of magnitude for the work function. It is worth noting, however, that the smaller internal potential of potassium as compared with the values for the other metals, which are all of similar size, is consistent with the greater number of conducting electrons in this metal. Nevertheless, the work function of potassium comes out far too large. Bismuth has been left out of account in the table. 

The minimum energy gap is also the important factor for other properties of a solid which depend on the electrons in the conduction band. These include the Pauli spin paramagnetism, and the (small) contribution of the electrons to thermal conductivity. All of these properties are due to extremely small concentrations of free electrons. Thus for silicon, where El = 1.1 eV, the number of conduction electrons is only 2 x 10 /cm, compared with an atom concentration of 5 X 10 /cm. This is for a sample where impurity concentrations have been reduced to 1 part in 10 by zone refining. 

C REDEFINE AS THE NUMBER OF CONDUCTION ELECTRONS FOR THE C NEUTRAL ATOM... 

The remainder of the data for SCFC is the standard output of the atomic programme RHFS, and consists essentially of the core- and the total atomic charge densities. The first line has 40 characters of text describing the atomic calculation followed by IDENT which must be CORE, the number of conduction electrons NEL, here 6, the first r value R1 on the radial mesh, and IFORM which determines the format to be used when reading the following... 

---