Jellium model surfaces

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.

The C, values for Sb faces are noticeably lower than those for Bi. Just as for Bi, the closest-packed faces show the lowest values of C, [except Bi(lll) and Sb(lll)].28,152,153 This result is in good agreement with the theory428,429 based on the jellium model for the metal and the simple hard sphere model for the electrolyte solution. The adsorption of organic compounds at Sb and Bi single-crystal face electrodes28,152,726 shows that the surface activity of Bi(lll) and Sb(lll) is lower than for the other planes. Thus the anomalous position of Sb(lll) as well as Bi(lll) is probably caused by a more pronounced influence of the capacitance of the metal phase compared with other Sb and Bi faces28... 

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

Since each atom in a f.c.c. array of purely metallic atoms is the same as every other atom (except at the surface), only a representative positive ion needs to be considered. Let it interact with a spherical portion (radius = R) of the electron gas which has a density of one electron per ion. This is called the jellium model. 

Figure 3.4 Distribution of the electronic density in the jellium model the metal occupies the region x < 0. The unmarked curve is for an uncharged surface, the other two curves are for the indicated surface-charge densities. The distance along the x axis is measured in atomic units (a.u.), where 1 a.u. of length = 0.529 A.

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. 

To explain where the surface contribution to the work function comes from, we need a model for the electron distribution in the surface region of a metal. One of the simplest is the jellium model [18]. 

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 some cases, macroscopic models are used for simplified discussions of certain phenomena without atomic resolution. A macroscopic tip-sample distance should be defined. To avoid confusion, we use the term barrier thickness instead. Throughout the book, the barrier thickness is always denoted by a upper-case letter, such as W or L. In the Sommerfeld model of the free-electron metals, the barrier thickness is the distance between the surface of the metal pieces. In the jellium model (see Chapter 4), the barrier thickness is defined as the distance between the image-force planes. 

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.

An early application of the jellium model is to estimate the work function (Bardeen, 1936 Smith, 1968). In the jellium model, there is only one parameter G. The work function is then a function of r, only. In reality, the work function depends not only on the material, but also on the crystallographic orientation of the surface. For most metals used in STM, the work function predicted by the jellium model is 1-2 eV smaller than the experimentally observed values, as shown in Table 4.1. 

Fig. 4.4. Surface energy predicted by the jellium model. A test case of the accuracy of the jellium model, conducted by Lang and Kohn (1970). Only for four alkali metals, Na, K, Rb, and Cs, are the predictions of surface energy by the jellium model fair. For most metals, with r, < 2.5 bohr, the surface energy predicted by the jellium model is negative, contradicting seriously with experimental facts. On the other hand, the calculated values of surface energies with crystal lattices agree much better with the values measured experimentally. (After Lang and Kohn, 1970).

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... 

We saw in Section 6.6.7 that the metal electrons seem to cany the main responsibility for the metal properties. One of the models that seems to be more successful in explaining the distribution of electrons on the metal surface and thus their interactions with solution, is the so-called jellium model. 

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 ... 

G. Gerber By applying two-photon ionization spectroscopy with tunable femtosecond laser pulses we recorded the absorption through intermediate resonances in cluster sizes Na with n = 3,. 21. The fragmentation channels and decay pattern vary not only for different cluster sizes but also for different resonances corresponding to a particular size n. This variation of r and the fragmentation channels cannot be explained by collective type processes (jellium model with surface plasmon excitation) but rather require molecular structure type calculations and considerations. 

Experiments show that the values for the double layer capacity at single crystal metal surfaces depend on the nature of the metal. This indicates that the metal surface cannot be considered as a perfect conductor, as was done in classical theories. It is well known that an overspill of metal electrons can occur at the boundary of a metal with vacuum. A similar overspill expected at a metal-solution interface would alter the double layer capacity by an amount depending on the type of metal. Models have been constructed in which the metal is represented by an electronic plasma in a uniformly, positively charged background, which is known as the jellium model [81,82]. The inclusion of the electron overspill into the integral equation theories of the electric double layer has been performed basically with the HAB model. 

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