Fermi level definition

The quasi-Fermi level is often interpreted as a thermodynamic driving force. Whether or not this is appropriate is a matter of some debate. While it may provide useful insights, the concept has been derived from kinetic arguments, and can at best provide a quasi-thermodynamic description. Whereas true equilibrium thermodynamics are universally valid, the predictive value of the quasi-Fermi level as a thermodynamic driving force for, e.g., chemical reactions may depend on the reaction mechanism. One particular aspect to be noted in this respect is that the quasi-Fermi level definitions in (2.62) and (2.63) only consider electrons and holes in the conduction and valence bands. They do not account for any changes in the... 

Equation (22) shows that since electrode potentials measure electronic energies, their zero level is the same as that for electronic energy. Equation (22) expresses the possibility of a comparison between electrochemical and UHV quantities. Since the definition of 0 is6 the minimum work to extract an electron from the Fermi level of a metal in a vacuum, the definition of electrode potential in the UHV scale is the minimum work to extract an electron from the Fermi level of a metal covered by a (macroscopic) layer of solvent. ... 

Cu, Ag, and Au are sd-metals (the d-band is complete but its top is not far from the Fermi level, with a possible influence on surface bond formation) and belong to the same group (I B) of the periodic table. Their scattered positions definitely rule out the possibility of making correlations within a group rather than within a period. Their AX values vary in the sequence Au < Ag < Cu and are quantitatively closer to that for Ga than for the sp-metals. This is especially the case ofCu. The values of AX have not been included in Table 27 since they will be discussed in connection with single-crystal faces. 

Figure 5.7. Schematic representation of the definitions of work function O, chemical potential of electrons i, electrochemical potential of electrons or Fermi level p = EF, surface potential %, Galvani (or inner) potential

It must be emphasized that Equations (5.24) and (5.25) stem from the definitions of Fermi level, work function and Volta potential and are generally valid for any electrochemical cell, solid state or aqueous. We can now compare these equations with the corresponding experimental equations (5.18) and (5.19) found to hold, under rather broad temperature, gaseous composition and overpotential conditions (Figs. 5.8 to 5.16), in solid state electrochemistry ... 

One might righteously ask why this close and preferential connection exists between the r vs and the r vs po dependencies. The answer is straightforward and has simply to do with the definitions of O and Fermi level EF (or electrochemical potential of electrons j (=EF))7 which are connected via ... 

Figure 6.14c shows the electron backdonation interaction (electrons are transferred from the Fermi level of the metal to the hybridized 27t molecular orbital which was originally empty, thus this is, by definition, a backdonation interaction). 

This, at first perhaps surprising fact, is important to remember as the same situation arises in solid state electrochemistry. To understand its validity it suffices to remember that the definition of the reference (zero) energy level of electrons for the she scale is simply the state of an electron at the Fermi level of any metal in equilibrium with an aqueous solution of pH=0 and pH2=l atm at 25°C. 

Figure 7.13. The definitions of ionization potential, Ie, work function, , Fermi level, EF, conduction level, Ec, valence level Ev, and x-potential Xe without (a) and with (b) band bending at the semiconductor-vacuum interface.

For electrons in a metal the work function is defined as the minimum work required to take an electron from inside the metal to a place just outside (c.f. the preceding definition of the outer potential). In taking the electron across the metal surface, work is done against the surface dipole potential x So the work function contains a surface term, and it may hence be different for different surfaces of a single crystal. The work function is the negative of the Fermi level, provided the reference point for the latter is chosen just outside the metal surface. If the reference point for the Fermi level is taken to be the vacuum level instead, then Ep = —, since an extra work —eoV> is required to take the electron from the vacuum level to the surface of the metal. The relations of the electrochemical potential to the work function and the Fermi level are important because one may want to relate electrochemical and solid-state properties. 

The equation above is written using the units h c 1. The quantity y is a vector of Dirac matrices, m is the electron mass multiplied by a Dirac matrix. Fermi level. With this definition the energy functional is... 

ECb. Evb. Ef. ancl Eg are, respectively, the energies of the conduction band, of the valence band, of the Fermi level, and of the band gap. R and O stand for the reduced and oxidized species, respectively, of a redox couple in the electrolyte. Note, that the redox system is characterized by its standard potential referred to the normal hydrogen electrode (NHE) as a reference point, E°(nhe) (V) (right scale in Fig. 10.6a), while for solids the vacuum level is commonly used as a reference point, E(vac) (eV) (left scale in Fig. 10.6a). Note, that the energy and the potential-scale differ by the Faraday constant, F, E(vac) = F x E°(nhe). where F = 96 484.56 C/mol = 1.60219 10"19 C per electron, which is by definition 1e. The values of the two scales differ by about 4.5 eV, i.e., E(vac) = eE°(NHE) -4-5 eV, which corresponds to the energy required to bring an electron from the hydrogen electrode to the vacuum level. 

According to a proposed definition, the electron work function

Fermi level of the metal across a surface carrying no net charge, and to transfer it to infinity in a vacuum. The work function for polycrystalline metals cannot be precisely determined because it depends on the surface structure it is different for smooth and rough surfaces, and for different... 

The first effect is a change in the work function of the semiconductor. In Fig. 22 the work function is denoted by distance between the Fermi level and the level corresponding to the value of the potential outside the crystal. (We have in mind the thermoelectron work function which figures in Richardson s formula.) We have... 

Suppose we have a slab of semiconductor of thickness L placed in an external homogeneous transverse electric field. In this case the electron concentration on one of the surfaces will be increased as compared with the case of no field i.e., the Fermi level will be raised. On the opposite surface, on the contrary, the electron concentration will be reduced i.e., the Fermi level will be lowered. This is illustrated in Fig. 25, where the continuous lines indicate the band edges in the absence of the field, and the dashed lines indicate the band edges in the presence of the field. (To be definite, the surfaces are assumed to be negatively charged the field in Fig. 25 is directed from left to right.)... 

Fermi-level DOS 115 Jellium model 92—97 failures 97 schematic 94 surface energy 96 surface potential 93 work function 96 Johnson noise 252 Kohn-Sham equations 113 Kronig-Penney model 99 Laplace transforms 261, 262, 377 and feedback circuits 262 definition 261 short table 377 Lateral resolution... 

The detection of sharp plasmon absorption signifies the onset of metallic character. This phenomenon occurs in the presence of a conduction band intersected by the Fermi level, which enables electron-hole pairs of all energies, no matter how small, to be excited. A metal, of course, conducts current electrically and its resistivity has a positive temperature coefficient. On the basis of these definitions, aqueous 5-10 nm colloidal silver particles, in the millimolar concentration range, can be considered to be metallic. Smaller particles in the 100-A > D > 20-A size domain, which exhibit absorption spectra blue-shifted from the plasmon band (Fig. 80), have been suggested to be quasi-metallic [513] these particles are size-quantized [8-11]. Still smaller particles, having distinct absorption bands in the ultraviolet region, are non-metallic silver clusters. 

It s important to know how many electrons one has in one s molecule. Fe(II) has a different chemistry from Fe(III), and CR3+ carbocations are different from CRj radicals and CR3 anions. In the case of Re2Cl82, the archetypical quadruple bond, we have formally Re(III), d4, i.e., a total of eight electrons to put into the frontier orbitals of the dimer level scheme, 17. They fill the a, two x, and the 6 level for the explicit quadruple bond. What about the [PtHj2] polymer 12 Each monomer is d8. If there are Avogadro s number of unit cells, there will be Avogadro s number of levels in each bond. And each level has a place for two electrons. So the first four bands are filled, the xy, xz, yz, z2 bands. The Fermi level, the highest occupied molecular orbital (HOMO), is at the very top of the z2 band. (Strictly speaking, there is another thermodynamic definition of the Fermi level, appropriate both to metals and semiconductors,9 but here we will use the simple equivalence of the Fermi level with the HOMO.)... 

Metals and semiconductors have positive and negative slopes in their electrical resistivity p) vs. temperature (T) curves as schematically shown in (109) and (110), respectively. By definition, the Fermi surface disappears when a band gap opens at the Fermi level. If the Fermi surface nesting is complete, all the Fermi surface is removed by the appropriate orbital mixing. However, if the Fermi surface nesting is incomplete, only the nested portion of the surface is removed by orbital mixing. The unnested portion is left as small Fermi surface pockets. The system will thus retain its metallic properties although the number of carriers (i.e. those electrons at the Fermi level) will be... 

In the theory of non-equilibrium processes at solid state junction and also semiconductor-liquid interfaces, as developed in the previous section, frequently quasi-Fermi levels have been used for the description of minority carrier reactions [90, 91], A concept for a quantitative analysis for reactions at n- and p-type electrodes has been derived [92, 93], using the usual definition of a quasi-Fermi level (Eqs. (3a) and (3b)). Taking a valence band process as an example, the quasi-Fermi level concept can be illustrated as follows ... 

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