6.1The electron gas: work function, surface structure and energy

Distance z (Fermi wavelengths)

Figure 6.1. Energy diagram de®ning the terms f, DV, mÅ and the eVective potential VeV (z) in relation to the Fermi level and the bottom of the conduction band of a metal. This diagram is drawn to scale from the data in table 1 of Lang & Kohn (1970) for rs54. See text for discussion.

From this simple manipulation we can understand the following points: (1) mÅ is a bulk property, determined by the kinetic energy and exchange-correlation energy of the electron gas; (2) the fact that the work function f depends on the surface face {hkl} means that DV has to be a surface property also. This has various consequences, which are spelled out in the next sections; but ®rst, we need a bit of background theory. The details can be quite complicated, especially considering that there are (at least) two length scales in the problem, one connected with the electron gas, and another connected with the lattice of ions.

It is a good idea to understand the elements of density functional theory (DFT), even if only in outline, in the form that Lang and Kohn used in the early 1970s to derive values for the work function and surface energies of monovalent metals (Lang & Kohn 1970, 1971; Lang 1973). In order not to lose the thrust of the argument, this material is relegated to Appendix J. These calculations characterize free electron metals in general in terms of the radius (rs) which contains one electron; in particular, their calculations spanned the range 2,rs,6 (in units of the Bohr radius a0) which includes the alkali metals Li to Cs. Figure 6.1 is drawn to scale for rs54, which is close to the value needed to describe sodium.

186 6 Electronic structure and emission processes

Table 6.1. The work function of jellium and its components. Columns 2 and 3 represent the kinetic, and exchange-correlation energy respectively (after Lang & Kohn, 1971)

The main aspect of this model is the replacement of the (insoluble) many electron N-body problem by N one-electron problems with an eVective potential, VeV in ®gure 6.1, which is a functional of the electron density. This potential contains the original electron±nuclei and electron±electron terms, and also has a term to describe exchange and correlation between electrons. These terms have been worked out precisely for a uniform electron gas, corresponding to the interior of jellium, so that explicit, numerical values can be given to these energies as a function of electron density. The trick now is to apply these same numerical recipes to non-uniform densities, whence the term local density approximation (LDA). There are many further methods which try to correct LDA for non-local eVects and density gradients, such as the generalized gradient approximation (GGA), but it is not clear that they always produce a better result. In any case, we are now getting into the realm of arguments between specialists.

Some results of Lang & Kohn's work on jellium are indicated on ®gures 6.1 to 6.3. The electron density (®gure 6.2(a)), electrostatic potential and eVective potential (®gure 6.1) have oscillations normal to the surface in the self-consistent solution obtained; there are substantial cancellations between the various terms. The work function of these model alkali metals (®gure 6.3) varies weakly from Li (rs about 3.3) to Cs (rs about 5.6), whereas the individual components of the work function vary quite a lot. This model was the ®rst to get the order of magnitude, and the trends with rs correct: a big achievement. Note that the position of the ions do not enter this model at all: everything is due to the electron gas, and the importance of the exchange-correlation term mxc, and the variation of the electrostatic contribution, are evident in table 6.1.

In the quarter century since Lang & Kohn's initial work, there have been major developments within the jellium model. As computers have improved, this method has also been applied to clusters, especially of alkali metals, of increasing size. Figure 6.2(b) shows the comparison of the electron density in a spherical sodium atom cluster of more than 2500 atoms, modeled as jellium, compared with the free planar jellium surface on the same scale (Brack 1993). The only diVerence of note between the two curves is that the oscillations in the cluster produce a standing wave pattern at the center of the cluster, whereas they die away from the planar surface. This central peak

188 6 Electronic structure and emission processes

radius, rs (a.u.)

Figure 6.3. Work functions in the jellium model (full squares, Lang & Kohn 1971), compared with experimental data for polycrystalline alkali and alkaline earth metals (open circles: Michaelson 1977). The elements plotted are after Lang (1973) and the solid line fourth-order polynomial ®t to these points has been added.

(or dip) varies with electron energy and is dominated by the highest occupied states which vary with the exact cluster size, whereas the oscillations close to the surface are independent of such details.

The oscillations in the electron density are called Friedel oscillations; these occur when a more or less localized change in the positive charge density (the discontinuity at the jellium model surface being an extreme case) is coupled with a sharp Fermi surface. In other words, they are a feature of defects in metals in general, not just surfaces, and are an expression of Lindhard screening, which is screening in the high electron density limit. Screening in metals is so eVective that there are ripples in the response, corresponding to overscreening.

Recently, these electron density oscillations have been seen dramatically in STM images both of surface steps, and of individual adsorbed atoms on surfaces, reported in several papers from Eigler's IBM group. By assembling adatoms at low temperature into particular shapes, these `quantum corrals' can produce stationary waves of electron density on the surface which are sampled by the STM tip, and the corresponding Friedel oscillations are energy dependent; two examples from a circular assembly of 60 Fe atoms on Cu(001) are shown in ®gure 6.4.

Whether or not these eVects can be explained in detail as yet (Fe and Cu are both

Figure 6.4. A `quantum corral' of 60 Fe atoms assembled and viewed on Cu(001) by STM at 4K. The tip imaging parameters are (a) Vt 5110 mV and (b) 2 10 mV, with current I51 nA (after Crommie et al. 1995, reproduced with permission).

Figure 6.5. (a) Cross section of the free electron Fermi surface, radius kF; (b) the combination of traveling wave states 6k near a surface. See text for discussion.

transition metals with important d-bands), these oscillations are present in free electron theory. To see how such eVects arise, one needs to do as simple a calculation as possible, and try to understand how the physics interacts with the mathematics. The calculation done by Lang & Kohn goes roughly as follows, using ®gure 6.5 as a guide.

Consider pairs of states, ordered by their k-vector perpendicular to the surface, k and 2 k. Their wavefunction is c ,ck(z) exp i(kxx1ky y), and when 6k are combined to vanish in the vacuum (outside the surface), ck(z),sin(kz -gF), where gF is a phase factor, dependent on kF, since the origin doesn't have to be exactly at z50. Draw a Fermi sphere, radius kF, with the k-axis (perpendicular to the surface) as a unique axis, as in ®gure 6.5. Make a slice at k, dk thick; the density of states g(k) is just the area of this slice which is p(kF2 2 k2). Now we can write

where the limits of integration are 0 and kF, and with a bit of manipulation you should get the result