The image force

In this section, we discuss the concept of classical image force. The validity of this concept has been verified using quantum mechanics in a many-body formalism (Bardeen, 1936 Lang and Kohn, 1970 Appelbaum and Hamann, 1972 Herring, 1992). We will present it in Chapter 4. 

the vacuum potential is chosen to be zero — when the electron is far from the metal. 

When an electron is located inside a tunneling barrier, positive charges are induced on both metal surfaces. The total force can be treated by a series of image charges induced by both metal surfaces. The force acting on the electron inside the barrier is the sum of forces from all the image charges, as shown in Fig. 2.3, 

The potential can be obtained by integration. A convenient zero point is chosen by the condition that when W z, only the left-hand surface is effective, and the effect of the right-hand metal surface should disappear. We obtain (Simmons, 1969) 

The potential is symmetric with respect to both metal surfaces. In fact, by direct substitution, it is easy to show that the potential at W — z equals that at z. At the middle of the barrier, z = W/2, the potential is 

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As previously noted the work function O is the work required to bring an electron from the Fermi level of the metal to a point outside the metal where the image forces are negligible, i.e., typically 1 to 0.1 pm outside the metal surface.9,10,16 22 The Volta potential at this point is defined so that the energy required to bring an electron from that point to an "infinite" distance from the metal surface is e. ... 

It follows from Eq. (9.2) that the work function has a chemical and an electrostatic component. Its overall value can be measured, whereas an exact determination of its individual components is not possible. The chemical component depends on the interaction between the charge and the surrounding medium moreover, it includes the work performed in overcoming the image forces. 

In addition to the nonelectrostatic adsorptive force, there is an image force between a dipole and a metal, which will be present whenever charged or dipolar particles in a medium of one dielectric constant are near a region of another dielectric constant. If the metal is treated as an ideal conductor, the image-force contribution to the energy of a dipole in the electrolyte is proportional to p2j z3, where z is the distance of the dipole from the plane boundary of the metal (considered ideal, with no surface structure), and to 1 + cos2 0. This ideal term is, of course, the same for all metals. If... 

The interface is, from a general point of view, an inhomogeneous dielectric medium. The effects of a dielectric permittivity, which need not be local and which varies in space, on the distribution of charged particles (ions of the electrolyte), were analyzed and discussed briefly by Vorotyntsev.78 Simple models for the system include, in addition to the image-force interaction, a potential representing interaction of ions with the metal electrons. 

The dielectric displacement must be calculated from electrostatics for a reactant in front of a metal surface the image force has to be considered. For the simple case of a spherical ion in front of a metal electrode experiencing the full image interaction, a straightforward calculation gives ... 

Similar derivations apply for the electrochemical case when taking the image force effect into account. More precisely, in the expression of the potential at the surface of the A sphere, the contribution of the electrical image of A (which bears an opposite charge and which center is located at a distance d, from the center of A) has to be taken into account. 

Fig. 1.38. The Fowler Nordheim equation for field emission. The relevant dimensions are much smaller than the tip radius. Therefore, a one-dimensional model is adequate. Neglecting the image-force effect, the potential U(z) outside the metal surface is linear with respect to distance z- The relevant parameters are work function of the material, tj), and the field intensity near the surface, F.

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. 

Fig. 2.7. Apparent barrier height calculated from the exact solution. Variation of the apparent barrier height 0.95(

In terms of A and eV, K 0.51- y(A, q = 0.51- /. The result of a direct numerical calculation is illustrated in Fig. 2.7. As shown, the apparent barrier height is almost a constant up to the point the barrier collapses. Actually, when the barrier is very thin, a square barrier following Eq. (2.15) becomes a poor approximation for the effect of the image force. The actual barrier is more slim than the square barrier, and the current is growing even faster. The apparent barrier height should not drop, as in the left edge of Fig. 2.7. Rather, it remains almost constant, as also shown by numerical calculations of Teague (1987) (see Fig. 1.40). 

In this section, we will derive the correction factor for the tunneling current due to the image force. For a free metal surface, the image potential pertinent to this surface, Eq. (2.2), is always prc.sent. The the simple image potential is always an essential part of the free sample, and is always implied in any first-... 

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. 

Binnig, G., Garcia, N., Rohrer, H., Soler, J. M., and Flores, F. (1984). Electron-metal-surface interaction potential with vacuum tunneling Observation of the image force. Phys. Rev. B 30, 4816-4818. 

Two additional electrostatic forces, the space charge effect and the image force, are also present. As shown by Shapiro et al. [54], their effect can, however, be neglected when q and the dust concentration are sufficiently small. 

Thus, the image force acting on an unit charge q is... 

Consider that the point P is a point in vacuum and just outside the reach of the image-force interactions arising from the presence of the electrode. Then the work... 

Fig. 6.38. The outer, /M potential of the electrode is the work done to bring a unit of positive charge from infinity to a point Pjust outside the reach of the image forces from the electrode.

Fig. 6.43. The two stages of getting the inner, or < >, potential (a) The work done to bring a unit of positive test charge from infinity to a point just outside the range of the image forces defines the outer, or j/, potential, (b) The charge on the solution is then removed, and the solution is wrapped in an oriented-dipole layer. The work done to transport the test charge across the oriented-dipole layer defines the surface, or, %, potential. Thus, the total work to bring the test charge from infinity to a point just inside the solution is given by s = /s + xs.

The image forces are those induced by the appearance of the fictitious charge on the metal (image charge) created by the ion s charge. Similarly to the adsorbed water molecules (Section 6.7.2), ions also experience dispersion forces due to the induction of instantaneous fluctuations in the electron density clouds of continuous atoms—the adsorbed ion and the metal atom. Both these forces are of attractive character and were discussed in Section 6.7.2. 

One contribution is independent of the concentration of the electron defects, containing merely the chemical portion of the work to transfer one electron to infinity. We shall call this El. This term includes also the image-force effect. 

Figure 3. Dependence of Ni ion yield on azimuthal angle at various pol r angle for clean Ni(001) bombarded by 1500 eV Ar ions at normal incidence. The solid curves represent experimental data while the dashed curves are predicted values obtained by correcting the calculated yields for 1000 eV Ar ion bombardment for the presence of the image force.

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