Uniform density of states

Conductivity at non-zero frequency is not constrained by the requirement that carriers must have a conducting path completely through the material. Hopping back and forth between two localized states contributes to the ac conductivity ), but makes no contribution to the dc conductivity. Consequently ct((o) is larger than a(0) and is often dominated by hopping between pairs of states. The conductivity due to hopping near E in a uniform density of states is (Austin and Mott 1969). 

For the approximate, uniform density of states 10/lTj, the formation of a moment becomes an all-or-nothing proposition, called strong ferromagnetism. Clearly if the density of states were nonuniform, the energy might be minimized for an intermediate shift of the bands, called weak ferromagnetism. We shall. see that both occur. 

Exchange interaction,, in cV, for the transition metals, determined from experimental optical spectra (Moore, 1949, 1952), as indicated in the text. Also listed is the critical value of l/ = WJ5, above which the uniform-density-of-states model predicts ferromagnetism H /5 was obtained from the Solid State Table. 

Fig. 61. Magnitude of DLTS electron emission peak versus bias pulse amplitude for sample JH139 from data similar to that in Fig. 60. The ambient reverse bias was 5 V. The data represented by O and a correspond to voltage pulse durations of i and of the DLTS emission time constant, respectively. The solid line was calculated assuming a spatially uniform density of states. Note the good agreement between theory and experiment for the longer pulse data. [From Lang et al. (1982a).]...

Electronic conduction in insulating materials has been a subject of considerable interest in the quest to understand charge transport in the thin film layers of organic electronic devices. In typical dielectric materials, the electronic states near the Fermi level are usually localized states, and the electron wave functions decay exponentially over a distance known as the localization length. In constrast, metals have a high, generally uniform density of states, whereas semiconductors have well-separated conduction and valence bands (separated... 

The previous section considered the case in which the characteristics of charge transport were defined by isotropic charge transport and a spatially uniform density of states function. These assumptions may not always be appropriate for OPVs. 

Equation 5 represents the theoretical prediction of the tunneling model for the time and temperature-dependent broadening of spectral holes. This result, however, is the result of a particular form of the density of tunneling states P E, R), which is based on the a priori assumption of Eq. 2. A uniform density of states in A is a physically reasonable choice the independence of 1, however, is difficult to justify, since k consists of several parameters [28]. The temperature dependence of spectral diffusion is dominated by A the time evolution stems mainly from k. Therefore, these two predictions from Eq. 5 do not have the same validity. In our experiments we have investigated time and temperature dependence separately. 

Additionally, for a given polymer with a uniform density of states there should be a linear relationship between the charge transferred from a series of metals and their work function. Experimentation confirms some of these predictions. 

Figure 5.18. Schematic representation of the density of states N(E) in the conduction band and of the definitions of work function d>, chemical potential of electrons p, electrochemical potential of electrons or Fermi level p, surface potential x> Galvani (or inner) potential

At x = 0, B(0) is equal to the uniform density of electrons. The first term of the right hand side makes a bulk peak around x = 0. It sharply damps outside, because the k-integration over the occupied states is similar in structure to the following damping oscillation function ... 

The state densities are not uniform across the energy band, and their population density, N(E), is the greatest in the center of the band. The number of electrons in the band, Ug, can then be evaluated and used in Eq. (6.9) by integrating the product of the density of state N(E) and the probability of their occupation, /( ), over the band energy range (see Figure 6.3c). Thus, for metals. 

The transition-metal density of states n(E) is not uniform throughout the band, as shown schematically in Fig. 7.4 but displays considerable structure that is characteristic of the given crystal lattice. This is seen in.Fig. 7.6,... 

The assumption that a is proportional to the square of the density of states is normally deduced only in the regime Edwards cancellation theorem. As shown in Chapter 1, Section 6.3, however, this is valid only if we can write vf=h l dE/dk and deduce vf ccg 1, which means a uniform expansion of the band. This is certainly not so for weak localization. For the singularity predicted by Altshuler and Aronov, any correction to dE/dk will be zero if averaged over a range kBTabout EF. We consider then that if a>%,... 

The manifold of the zero-order levels has uniformly spaced levels with a separation e = p 1, where p is the (constant) density of states. The zero-order energies are... 

The failure is not limited to metal-ammonia solutions nor to the linear Thomas-Fermi theory (19). The metals physicist has known for 30 years that the theory of electron interactions is unsatisfactory. E. Wigner showed in 1934 that a dilute electron gas (in the presence of a uniform positive charge density) would condense into an electron crystal wherein the electrons occupy the fixed positions of a lattice. Weaker correlations doubtless exist in the present case and have not been properly treated as yet. Studies on metal-ammonia solutions may help resolve this problem. But one or another form of this problem—the inadequate understanding of electron correlations—precludes any conclusive theoretical treatment of the conductivity in terms of, say, effective mass at present. The effective mass may be introduced to account for errors in the density of states—not in the electron correlations. 

Gauss s law states that if an imaginary enclosed surface of area S is drawn around a uniform density of charge, the component of the electric field at the surface, which is perpendicular to the surface, is proportional to the total charge enclosed. Hence, Gauss s law is expressed as... 

The assumption about a uniform probability for any distribution of the energy between the harmonic oscillators may now be used to determine the probability Pet >e (E). It can be expressed as the ratio between the density of states corresponding to the situation where the energy exceeds the threshold energy in the reaction coordinate and the total density of states at energy E, that is, N(E) of Eq. (7.36). 

SS, TS) are logarithmic, due to time-inversion symmetry. In one dimension, the particle-hole susceptibilities (CDW, SDW) are also logarithmic, due to the property of nesting of the Fermi surface The left sheet of Fig. la superimposes on the right sheet in a translation by 2kF. The static, uniform magnetic susceptibility is the usual Pauli one measuring the density of states at the Fermi level ... 

Exactly how this is done is st en in Fig. 20-11, where the model density of states from Fig. 20-8,b is redrawn (rotated 90") with the free-clectron and r/-like densities shown. separately. For a metal in a particular Dn column of the Solid Stale Table (3 < J7 < II), t l should be picked. so that the integral over the occupied portions of the two densities of states shown—and given in Eqs. (20-10) and (20-11)—comes to n electrons. This is carried out in Problem 20-3 to obtain the number Z, of free electrons per atom. The results arc shown in Table 20-4. We shall. sec in Section 20-F that the magnetic properties suggest a value for the 3d series of Z, = 0.6. The discrepancy is not large in view of the assumption of a uniform density of d slates. 

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