Free electron metals

Figure 6.11. Schematic energy diagram of an atom approaching a free electron metal with an sp band. Notice that the vacuum level is the...

Figure 6.17. DOS diagrams showing schematically the electron density around the Fermi level for a free-electron metal, a transition metal, and an insulator.

For a material to be a good conductor it must be possible to excite an electron from the valence band (the states below the Fermi level) to the conduction band (an empty state above the Fermi level) in which it can move freely through the solid. The Pauli principle forbids this in a state below the Fermi level, where all states are occupied. In the free-electron metal of Fig. 6.14 there will be plenty of electrons in the conduction band at any nonzero temperature - just as there will be holes in the valence band - that can undertake the transport necessary for conduction. This is the case for metals such as sodium, potassium, calcium, magnesium and aluminium. 

The transition metals are also good conductors as they have a similar sp band as the free-electron metals, plus a partially filled d band. The Group IB metals, copper, silver and gold, represent borderline cases, as the d band is filled and located a few eV below the Fermi level. Their sp band, however, ensures that these metals are good conductors. 

Here we try to gain insight into the trends in reactivity of the metals without getting lost in too much detail. We therefore invoke rather crude approximations. The electronic structure of many metals shows numerous similarities with respect to the sp band, with the metals behaving essentially as free-electron metals. Variations in properties are due to the extent of filling of the d band. We completely neglect the lanthanides and actinides where a localized f orbital is filled, as these metals hardly play a role in catalysis. 

In this section we summarize the main results in simple and idealized schemes. We consider adsorption on a free electron metal, and on a transition metal. In particularly the adsorption of a molecule on a metal with d states is of great interest for catalysis. 

A free-electron metal only possesses a broad sp band. Upon approach, the electron levels of the adsorbate broaden and shift down in energy, implying that the adsorbate becomes more stable when adsorbed on the metal. The interaction results in a bonding energy of typically 5 eV for atomic adsorbates on metals. The situation is illustrated in Fig. 6.23. 

Figure 6.23. The energy levels of an adsorbate broaden and are lowered in energy when it approaches a free-electron metal with a broad sp band. Note that the initially empty upper...

When an atom with a filled level at energy approaches a metal surface it will first of all chemisorb due to the interaction with the sp electrons of the metal. Consider for example an oxygen atom. The 2p level contains four electrons when the atom is isolated, but as it approaches the metal the 2p levels broaden and shift down in energy through the interaction with the sp band of the metal. Fig. 6.28(a) and (b) show this for adsorption on jellium, the ideal free-electron metal. 

For a free-electron metal the refractive index as a function of the frequency oj is given by the following expression ... 

FIG. 7. The importance of the tip state is highlighted by applying the reciprocity principle to imaging a free electron metal surface with a d/ tip. (From Ref. 39.)... 

Fig. 1.4. A one-dimensional metal-vacuum-metal tunneling junction. The. sample, left, and the tip, right, are modeled as semi-infinite pieces of free-electron metal.

If in an STS experiment the goal is to obtain the DOS of the sample, one requires a tip with a constant DOS, or a free-electron metal tip. In this case, from Eq. (1.21),... 

Thus, with a free-electron metal tip, the dynamic tunneling conductance is proportional to the DOS of the sample. 

Fig. 1.23. Free-electron-metal model of STM resolution. The sample is modeled as a corrugated free-electron-metal surface. The tip is modeled as a curved free-electron-metal surface with radius r, at the closest approach to the sample surface. (After Stoll, 1984.)...

The original 5-wave-tip model described the tip as a macroscopic spherical potential well, for example, with r 9 A. It describes the protruded end of a free-electron-metal tip. Another incarnation of the 5-wave-tip model is the Na-atom-tip model. It assumes that the tip is an alkali metal atom, for example, a Na atom, weakly adsorbed on a metal surface (Lang, 1986 see Section 6.3). Similar to the original 5-wave model, the Na-atom-tip model predicts a very low intrinsic lateral resolution. 

Fig. 1.30. Origin of atomic resolution on metal surfaces. According to the reciprocity principle, the image taken with a d,- tip state (which exists on a W tip) on a free-electron-metal surface is equivalent to an image taken with a point tip on a fictitious sample surface with a d state on each top-layer atom, which obviously has a strong corrugation. (Reproduced from Chen, 1990, with permission.)...

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 metals is the Sommerfeld theory of free-electron metals (Ashcroft and Mermin 1985, Chapter 2), where a metal is described by a single parameter, the conduction electron density n. A widely used measure of... 

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... 

In the bulk, the charge density of electrons n equals in magnitude the charge density of the uniform positive charge background +, thus to preserve charge neutrality. The only parameter in the jellium model, r,, is the same as in the Sommerfeld theory of free-electron metals. 

The theory of field-emission spectroscopy for free-electron metals was developed by Young (1959). We present here a simplified version of Young s theory, which includes all the essential physics related to the experimental observation of surface states. 

Fig. 4.7. Field emission spectra of W(112) and W(IOO). Dotted curve theoretical field emission spectrum for free electron metals. Dashed curve experimental field emission spectrum for W(112). Solid curve experimental field emission spectrum for W(IOO), A substantial deviation from the free electron metal behavior is observed. The deviation, so-called Swanson hump, is due to the dominating role of localized surface states near the Fermi level at W(IOO) surface in field emission. (After Swanson and Grouser, 1967).

Fig. 4.8. Field-emission spectrum of Mo(lOO). The quantity displayed, Jf, is the ratio between the observed field-emission current and the prediction based on a free-electron model, Eq. (4.20). As shown, the field-emission spectrum of Mo(lOO) near the Fermi level is substantially different from a free-electron-metal behavior. (After Weng, 1977.)...

The first successful first-principle theoretical studies of the electronic structure of solid surfaces were conducted by Appelbaum and Hamann on Na (1972) and A1 (1973). Within a few years, first-principles calculations for a number of important materials, from nearly free-electron metals to f-band metals and semiconductors, were published, as summarized in the first review article by Appelbaum and Hamann (1976). Extensive reviews of the first-principles calculations for metal surfaces (Inglesfeld, 1982) and semiconductors (Lieske, 1984) are published. A current interest is the reconstruction of surfaces. Because of the refinement of the calculation of total energy of surfaces, tiny differences of the energies of different reconstructions can be assessed accurately. As examples, there are the study of bonding and reconstruction of the W(OOl) surface by Singh and Krakauer (1988), and the study of the surface reconstruction of Ag(llO) by Fu and Ho (1989). 

For free-electron metals, the local density of states near the Fermi level is proportional to the total valence-electron charge density. Therefore, up to an overall constant depending on the bias V, the tunneling current is proportional to the charge density at the nucleus of the apex atom ... 

On surfaces of some d band metals, the 4= states dominated the surface Fermi-level LDOS. Therefore, the corrugation of charge density near the Fermi level is much higher than that of free-electron metals. This fact has been verified by helium-beam diffraction experiments and theoretical calculations (Drakova, Doyen, and Trentini, 1985). If the tip state is also a d state, the corrugation amplitude can be two orders of magnitude greater than the predictions of the 4-wave tip theory, Eq. (1.27) (Tersoff and Hamann, 1985). The maximum enhancement factor, when both the surface and the tip have d- states, can be calculated from the last row of Table 6.2. For Pt(lll), the lattice constant is 2.79 A, and b = 2.60 A . The value of the work function is c() w 4 cV, and k 1.02 A . From Eq. (6.54), y 3.31 A . The enhancement factor is... 

The modification of an x-wave sample state due to the existence of the tip is similar to the case of the hydrogen molecule ion. For nearly free-electron metals, the surface electron density can be considered as the superposition of the x-wave electron densities of individual atoms. In the presence of an exotic atom, the tip, the electron density of each atom is multiplied by a numerical constant, 4/e 1.472. Therefore, the total density of the valence electron of the metal surface in the gap is multiplied by the same constant, 1.472. Consequently, the corrugation amplitude remains unchanged. 

As we showed in the previous section, in order to obtain reproducible tunneling spectra, the STM tip must have reproducible DOS, preferably a flat DOS, that is, with a free-electron-metal behavior. However, the tips freshly made by mechanical or electrochemical methods, especially those providing atomic resolution, often show nonreproducible tunneling spectra. The DOS of such tips is often highly structured. To obtain reproducible STS data, a special and reproducible tip treatment procedure is required. 

---