Electron density states

Fig. 1-2. Energy distribution of electrons near the Fermi level, cf> in metal crystals c = electron energy f(.i) s distribution function (probability density) ZXe) = electron state density, = occupied...

Fig. 2- State density distribution curve of electrons in solid ZXe) = electron state density tu = uiq>er band edge = lower band edge.

Electron occupation in the frontier bands of metal crystals varies with different metals as shown in Fig. 2-7. For metallic iron the frontier bands consist of hybridized 4s-3d-4p orbitals, in which 4s and 3d are partially occupied by electrons but 4p is vacant for electrons. Figure 2-8 shows the electron state density curve of metallic iron, where the 3d and 4s bands are partially filled with electrons. Electrons in metals occupy the energy states in a frontier band successively fix>m the lower band edge level to the Fermi level, leaving the higher levels vacant. 

Fig. 2-13. Schematic electron state density distribution curves in the valence and conduction bands of silicon cc = conduction band edge level Cv = valence band edge level c, = band gap (1.1 eV for silicon) CB = conduction band V6 = valence band.

Fig. 2-16. Electron state density distribution and electron-hole pair formation in the conduction and valence bands of intrinsic semiconductors Cf > Fermi level of intrinsic semiconductors.

Fig. 2-20. Electron state density and ranges of Fermi energy where electron occupation probability in the conduction band of an electron ensemble of low electron density (e.g., semiconductor) follows Boltzmann function (Y i)or Fermi function (y > 1) y = electron activity coeffident ET =transition level from Y 4= 1 to Y > 1 0(t) = electron energy state density CB = conduction band. [From Rosenberg, I960.]...

The semiconductor surface where the Fermi level is pinned at a surface state of high density (Fig. 2-31) is in the state of degeneracy of electron levels, because of the high electron state density at the surface Fermi level. Similarly, the surface degeneracy is also established when the band bending becomes so great that the Fermi level is pinned either in the conduction band or in the valence band as shown in Fig. 2-32. 

Since the electron state density near the Fermi level at the degenerated surface (Fermi level pinning) is so high as to be comparable with that of metals, the Fermi level pinning at the surface state, at the conduction band, or at the valence band, is often called the quasi-metallization of semiconductor surfaces. As is described in Chap. 8, the quasi-metallized surface occasionally plays an important role in semiconductor electrode reactions. 

The electron state densities DredCe) and Z ox(e) in the donor and acceptor bands of hydrated redox particles are given by the product of the probability densities Wrbd(c) and Wcacie) and the concentrations Nkbd and Nox, respectvely, in Eqns. 2-48 and 2-49 ... 

Fig. 2-40. Distribution of electron state density of hydrated redox particles (a) oxidant concentration JVox equal to reductant concentrationNRED. (b) oxidant concentration iVox greater than reductant concentration NgEo cnsEDox) = Fermi level of redox electrons.

Similarly, the total electron state density in the metal electrode is expressed by the sum of the state densities of occupied electrons and vacant electrons as shown in Eqn. 8-4 ... 

Figure 8-3 shows an energy diagram of the total electron state density Aiedox(s) of redox particles, the occupied electron state density Aied(e) of reductant particles, and the unoccupied electron state density Dca t) of oxidant particles as functions of electron energy e. 

Fig. 8-3. Electron state density in hydrated reductant and oxidant particles near the Fermi level of redox electrons I>redox = 1)red -I>ox= electron state density in redox particles Dpcredox) 1)f(bed)l nox) = electron state density in redox particles at the Fermi level of redox electrons.

Fig. 8-4. (a) Electron state density D in a metal electrode and in hydrated redox particles, (b) rate constant for electron ttmneling k, and (c) exchange reaction current electron transfer in eqiiilibrium with a redox reaction sl = lower edge of an allowed band of metal electrons. [From Gerischer, I960.]... 

Figures 8-5 and 8-6 are energy diagrams, as functions of electron energy e imder anodic and cathodic polarization, respectively, for the electron state density Dyf.t) in the metal electrode the electron state density AtEDox(c) in the redox particles and the differential reaction current ((e). From these figures it is revealed that most of the reaction current of redox electron transfer occurs in a narrow range of energy centered at the Fermi level of metal electrode even in the state of polarization. Further, polarization of the electrode potential causes the ratio to change between the occupied electron state density Dazc/itnu md the imoccupied...

Figure 8-11 shows as a function of electron energy e the electron state density Dgdit) in semiconductor electrodes, and the electron state density Z e) in metal electrodes. Both Dsd.t) and AKe) are in the state of electron transfer equilibrium with the state density Z>bei)ox(c) of hydrated redox particles the Fermi level is equilibrated between the redox particles and the electrode. For metal electrodes the electron state density Ai(e) is high at the Fermi level, and most of the electron transfer current occurs at the Fermi level enio. For semiconductor electrodes the Fermi level enao is located in the band gap where no electron level is available for the electron transfer (I>sc(ef(so) = 0) and, hence, no electron transfer current can occur at the Fermi level erso. Electron transfer is allowed to occur only within the conduction and valence bands where the state density of electrons is high (Dsc(e) > 0). 

Fig. 8-11. Electron state density in a metal electrode, semiconductor electrode, and redox particles in equilibrium with a redox electron transfer reaction. [From Glerischer, 1961.]...

Electron state density in redox electrode reactions... 

Fig. 8-14. Electron state density for a redox electron transfer reaction and profile of electrostatic inner potential, across an electrode interface = potential...

Fig. 8-16. Electron state density for a redox electron transfer reaction of h3rdrated redox particles at semiconductor electrodes (a) in the state of band edge level pinning and (b) in the state of Fermi level pinning dashed curve = band edge levels in reaction equilibrium solid curve = band edge levels in anodic polarization e p,sq = Fermi level of electrode in anodic polarization e v and c c = band edge levels in anodic polarization.

Fig. 8-16. Electron state density in a semiconductor electrode and in hjrdrated redox partides, rate constant of electron tunneling, and exchange redox current in equilibrium with a redox electron transfer reaction for which the Fermi level is close to the conduction band edge eF(sc) = Fermi level of intrinsic semiconductor at the flat band potential 1. 0 (tp.o) = exchange reaction current of electrons (holes) (hvp)) - tunneling rate constant of electrons (holes).

Fig. 8-39. Electron state density in an electrode metal, Du, a semiconductor film, Dt, hydrated redox particles, Dredox, and exchange reaction current of redox electrons, t., in electron transfer equilibrium M = exchange current at a bare metal electrode, M/F= exchange current at a thin-film-covered metal electrode.

Fig. 9-19. Electron state density of adsorbed proton/lpndrogen redox particles on metal electrodes (a) the relative concentration of adsorbed reductant hydrogen atoms (Had) will be hitler if the Fermi level ef(h, of electrode is higher than the standard Fermi level of adsorbed redox particles, (b) the relative concentration of adsorbed oxidant...

Fig. 9-20. Electron state density D of adsorbed redox particles ) on semiconductor...

An electronic effect due to the variation of the electronic states density of the metal atoms in the vicinity of sulfur atoms. 

Increase of the electronic state density in the carbon pore walls with the voltage. Hahn et al. [52] have measured double-layer capacitance and electronic conductance of an activated carbon electrode in an aprotic electrolyte solution, 1 mol/dm3 (C2H5)4NBF4 in acetonitrile. Both quantities show a similar dependency on the electrode potential with distinct minima near the potential of zero charge. This correlation suggests that the capacitance, like the conductance, is governed substantially by the electronic properties of the solid, rather than by the ionic properties of the solution in the interface of the double layer. 

This view of superconductivity in fullerenes in which T, is dominated by a simple volume-dependent electronic state density suggests that models [17] requiring pairing to be mediated by intermolecular vibrational modes that... 

Accounting high values of electron state densities on the Fermi level, we accepted that the ionization process for the surface of binary nanoclusters has adiabatie eharacter (Bom-Oppenheimer approximation) and it is described by a smooth potential curve of transition from the initial state (atom in nanoeluster) to final (ion inMe" solution) [47] (Fig. 2). 

---