Potentials Jellium model

Figure 2. Sketch of an uncharged metal surface (simulated by the jellium model) covered by a macroscopic solvent layer, showing the components of the electric potential drop. 8%M is the surface potential of the metal modified by the solvent layer %s + 6%s is the surface potential of the solvent modified by the contact with the metal %s is the unmodified surface potential of the solvent layer at the external surface.

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

In order to estimate the magnitude of the surface dipole potential and its variation with the charge density, we require a detailed model of the metal. Here we will explore the jellium model further, which was briefly mentioned in Chapter 3. 

Fig. 2-10. Profile of electron density and electronic potential energy across a metal/vacuum interface calculated by using the jellium model of metals MS = jellium surface of metals Xf = Fermi wave length p. - average positive charge density P- s negative charge density V = electron exchange and correlation energy V, - kinetic energy of electrons. [From Lange-Kohn, 1970.]...

In calculating the distribution of image charge aj.x), we use an improved jellium model in which the quasi-potential due to the lattice ions is taken into... 

The simplest model of a metal surface is the jellium model, which is a Sommerfeld metal with an abrupt boundary. In provides a useful semiquanti-tative description of the work function and the surface potential (Bardeen, 1936). It validates the independent-electron picture of surface electronic structure Essentially all the quantum mechanical many-body effects can be represented by the classical image force, which has been discussed briefly in Section... 

The jellium model for the surface electronic structure of free-electron metals was introduced by Bardeen (1936) for a treatment of the surface potential. In the jellium model, the lattice of positively charged cores is replaced by a uniform positive charge background, which drops abruptly to zero at the... 

Fig. 4.2. Charge distribution and surface potential in a jellium model, (a) Distribution of the positive charge (a uniform background abruptly drops to zero at the boundary) and the negative charge density, determined by a self-consistent field calculation. (b) Potential energy as seen by an electron. By including all the many-body effects, including the exchange potential and the correlation potential, the classical image potential provides an adequate approximation. (After Bardeen, 1936 see Herring, 1992.)...

Fig. 4.3. Position of the image plane in the jellium model. The surface potential of an electron in the jellium model is calculated using the local-density approximation. By fitting the numerically calculated surface potential with the classical image potential, Eq. (4.7), the position of the image plane is obtained as a function of r, and z. The results show that the classical image potential is accurate down to about 3 bohrs from the boundary of the uniform positive charge background. For metals used in STM, r, 2 — 3 bohr, zo 0.9 bohr. (Reproduced from Appelbaum and Hamann, 1972, with permission.

Similar to the failures of the free-electron model of metals (Ashcroft and Mermin, 1985, Chapter 3), the fundamental deficiency of the jellium model consists in its total neglect of the atomic structure of the solids. Furthermore, because the jellium model does not have band structure, it does not support the concept of surface states. Regarding STM, the jellium model predicts the correct surface potential (the image force), and is useful for interpreting the distance dependence of tunneling current. However, it is inapplicable for describing STM images with atomic resolution. 

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... 

Jellium model of the metal, 890 and crystal structure, 892 and pseudo potentials, 892 and surface of potential, 893 Jeng, organic adsorption, 975, 979 Jovancicevic, 1125, 1263 Junction c-i, 1081... 

Through the jellium model of the metal we have explained the effect of the metal electrons on the interfacial properties. We also know that the spillover of electrons creates a separation of charges at the metal edge, and consequently, a surface potential. However, what is the magnitude of this surface potential How important is its contribution to the total potential drop in the interfacial region ... 

The KS exchange potential coefficient aKS x(0) is essentially the image-potential value of 1/4, ranging from 0.195 to 0.274 over the metallic range of densities. Its value is precisely 0.250 for 0 = yjl, which corresponds to a Wigner-Seitz radius of rg — 4.1. The jellium model is stable for approximately this value of rs. With the assumption that the asymptotic structure of vxc (r) is the image potential, we see that the correlation contribution to this structure is an... 

Fig. 3.13 Variation of the electrostatic potentials with distance from a metallic electrode, (a) Classical representation (b) The jellium model.

One of the earliest treatments of a metal surface was based upon a jellium model (Bardeen, 19.36). If the electron gas terminated abruptly at the surface of the jellium there would be no net potential to contain the electrons in the metal. Therefore the electron gas extends beyond the metal, giving a dipole layer, as illustrated in Fig. 17-5. Bardeen attempted the self-consistent calculation of the resulting potential. It should be mentioned that the Fermi-Thomas approximation is not adequate for this task and was not used by Bardeen it is not difficult to see that it would predict the Fermi energy to be at the vacuum level, corresponding to a vanishing work function. 

More recently, Lang and Kohn (1970) treated the jellium model with modern methods and modern computers. In particular, they included exchange terms in the frcc-clcctron approximation, as we have discussed. In lliis context it is interesting that the potential well that holds the electrons in the metal comes predominantly from theexchange potential the electrostatic potential itself gives only... 

Various refinements of the above model have been proposed for example, using alternative spherical potentials or allowing for nonspherical perturbations,and these can improve the agreement of the model with the abundance peaks observed in different experimental spectra. For small alkali metal clusters, the results are essentially equivalent to those obtained by TSH theory, for the simple reason that both approaches start from an assumption of zeroth-order spherical symmetry. This connection has been emphasized in two reviews,and also holds to some extent when considerations of symmetry breaking are applied. This aspect is discussed further below. The same shell structure is also observed in simple Hiickel calculations for alkali metals, again basically due to the symmetry of the systems considered. However, the developments of TSH theory, below, and the assumptions made in the jellium model itself, should make it clear that the latter approach is only likely to be successful for alkali and perhaps alkali earth metals. For example, recent results for aluminium clusters have led to the suggestion that symmetry-breaking effects are more important in these systems. ... 

---