Refs: Prutton, Chapter 4, pages 108-138; Zangwill, Chap 4, pages 54-109; M.C. Desjonqueres and D. Spanjaard 'Concepts in Surface Physics' (1993).
There are various consequences of this fact, which are spelled out in the next sections; but first, 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.
In the lecture I went through the outline of density functional theory, in the form that Lang and Kohn used to derive values for the workfunction and surface energies of monovalent metals in the early 70’s. I don’t feel like writing out these equations here, because it is not at all original, and you can better find them from the books and articles quoted. These calculations characterise the alkali metals in terms of the radius (rs) which contains 1 electron, with 2 < rs < 6 spanning the range of Li to Cs. This is also where Ashcroft and Mermin start from in their Chap 1.
The main aspect of this theory, is the replacement of the (insoluble) many electron N-body problem by a N one-electron problems with an effective potential, 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 the 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 is now to apply these same numerical recipes to non-uniform densities, whence the term Local Density Approximation or LDA. There are many further methods which try to correct LDA for non-local effects and density gradients, 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.
The results of Lang and Kohn’s work are indicated on diagrams A1-A6. The electron density (diagram A1), the electrostatic potential and the effective potential (diagrams A2, A4), all have oscillations normal to the surface when the self-consistent solution is obtained, and there are substantial cancellations between the various terms. You may note that the work function of these model alkali metals varies quite weakly from Li (rs about 3.3) to Cs (rs about 5.6) (diagram A5), whereas the individual components of the work function vary a lot (Table A3). This theory was the first to get the order of magnitude, and the trends with rs correct: a big achievement. Note that the position of the ions don’t figure in this model at all; it is all due to the electron gas.
The oscillations in the electron density are called Friedel Oscillations; these occur when any discontinuity in the positive charge density is coupled with a sharp Fermi Surface. In other words, they are a features 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 effective that there are ripples in the response, corresponding to overscreening. Recently, this has been seen dramatically in STM images of individual adsorbed atoms on surfaces, in several papers from Eigler's IBM group. By assembling adatoms at low T into particular shapes, these ‘Quantum Corrals’ can produce stationary waves of electron density on the surface which are sampled by the STM tip; you may wish to consult M.F. Crommie et al, Physica D83 (1995) 98-108, to see a variety of shapes and standing wave patterns for Fe atoms on Cu(100) at low T. Whether or not these effects can be explained in detail as yet, the oscillations are present in free electron theory. Clearly we also have a new profession here, the Quantum Cowboy: pure fun, pure physics and pure hype in equal proportions. It’s wonderful stuff!
To see how such effects arise, one needs to do as simple a calculation as possible, and try to see how the physics interacts with the maths. Ashcroft and Mermin, Chap 17 contains most of both, but I’m not sure that it is the easiest place to start. The calculation done by Lang and Kohn goes roughly as follows.
The various methods used for surfaces are the same as those you would come across in an advanced solid state course for studying band theory and related topics. However, there is no way, realistically, that experimentalists with a range of backgrounds, such as many of us, are going to become familiar with many of these methods at first hand. We can read the specialist articles and listen to talks at conferences. An increasing number of theorists are sufficiently practical and public-spirited that they collaborate closely with experimentalists, and make their codes available to others for work on specific problems. It is a welcome recent development that theorists have addressed the problem of ‘understanding’. By this I mean that they acknowledge that the ‘true’ solution is obtained by keeping all the terms in the Schrodinger equation that they can think of, but that this doesn’t necessarily help one understand trends in behaviour, or help one make predictions. Pettifor’s recent book, for example, starts with a quote from Einstein: ‘As simple as possible, but not simpler’. This is excellent: with such an attitude there is real prospect that we can ‘understand’ a higher proportion of theory than we would be able to otherwise.
To start with we need a few names of theoretical methods, for example pseudopotentials, OPW, APW, tight-binding, effective medium, etc. The older methods are described by Ashcroft and Mermin, and typically tight-binding (where interatomic overlap integrals are thought of as small, A and M, Chap 10) is thought of the opposite extreme to nearly-free electron theory (where Fourier coefficients of the lattice potential are thought of as small, A and M, Chap 9). However, this is more aparent than real, in that both pictures can work for arbitarily large overlap integrals or lattice potentials; the only requirement is that the basis sets are complete for the problem being studied. This is course can lead to some semantic problems: methods which sound different may not in fact be so different, and additional effects included are almost certainly not simply additive.
The basic feature caused by including the ions via any of these methods is that the electron density near the surface is now modulated in x and y with the periodicity of the lattice (diagram A8). However, there are now two length scales in the problem which compete; it is not required that surface states have oscillation periods which bear any relation to the lattice period in the z-direction (diagram A9). Many metal surface reconstructions are due to competing electronic effects of a quite complex kind; for example, Au(100) has a close packed (roughly 23 x 1) surface structure, in which the outer, almost hexagonal (111-like) layer, floats on top of the square (100) bulk structure. Explanations of the cohesive properties of Ag and Au need major departures from free electron theory, including the role of d-bands (see eg. Pettifor, Chap 7, pages 173-198); many theory groups quite reasonably fight shy of such complexity.
Increasingly what counts is the speed of the computer code; if this speed scales with a lower power of the number (N) of electrons in the system, then more complex/ larger problems can be tackled. Each major theory group essentially has its own code. Desjonqueres and Spanjaard probably have the most comprehensive description in book form of these methods in relation to surface problems. For example, Effective Medium Theory (EMT) calculations are fast enough that they can be used to simulate dynamic processes such as adsorption, nucleation or melting on metal surfaces; here a full electronic structure calculation is being done for each set of positions of the nuclei, i.e at each time step (see e.g. K. Jacobsen et al, Phys Rev B35 (1987) 7423; Pettifor, Chap 5.7, pages 131-135). This requires computer speeds that would have been inconceivable just a few years ago. Now one of the students in our class is downloading programs from the corresponding website in Denmark http://www.fysik.dtu.dk/camp-sd.html in order to run these programs as a class project!
There are several methods of measuring the work function, as described by Woodruff and Delchar, and by Holzl, amongst others. The work function varies with the surface face exposed, as shown for several elemental solids in Table A10. This variation gives rise to several interesting effects, as described here and in the next section.
A polycrystalline material, with different faces exposed, gives rise to fields outside the surface, referred to as patch fields. Such fields are very important for low energy electrons or ions in vacuum, and can thereby influence measurement accuracy in surface experiments. Molybdenum is often used for such critical parts of UHV apparatus, because the work function doesn’t vary more than 0.4 V between the low index faces (Table A10), whereas Nb and W, which are otherwise similar, have variations of around 0.8 V.
The origin of this face-specific nature of the work function can be seen qualitatively by considering ‘jellium’ again. First, draw the step function in the positive charge density and then think about how the electron charge density will respond to this step (diagram A11(a)). The negative charge spills over into the vacuum, causing a dipole layer, whose dipole moment is directed into the metal. Now use Gauss’ law and show that
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