Metal surfaces: jellium and the work functionPrepared as a project by Ben Saubi with John Venables,19981999 CPES, University of Sussex, Brighton, UK. 


Figure 1: Energy diagram defining the terms f, DV, mbar and the effective potentials V_{eff}(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 and Kohn (1970) for r_{s} = 4. See text for discussion.
We then note that E_{F}  V(¥ ) = mbar, the Fermi level with respect to the bottom of the conduction band, and that the work function, f = V(+¥ )  V(¥ )  mbar, or equivalently f + mbar = V(+¥ )  V(¥ ) º DV.
From this simple manipulation we can understand the following points:
Some results of Lang and Kohn’s work on jellium are indicated on figures given in the this project, and these can be shared in the lecture or looked up in your own time. The key point is that the electron density (figure 2a), the electrostatic potential and effective potential (figure 1) have oscillations normal to the surface in the selfconsistent solution obtained; there are substantial cancellations between the various terms.
(a) 

Figure 2a: Electron density at a metal surface in
the Jellium model, after Lang and Kohn (1970) for r_{s} = 2 and 5;
Figure 2b: Comparison between a spherical cluster
of 2654 simulated Na atoms (r_{s} = 3.96) and a planar
surface for r_{s} = 4 (after Genzken and Brack 1991 and Brack 1993).
In the quarter century since Lang and 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 2b 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 difference 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 (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.
To recap this section, the work function of these model alkali metals (figure 3) varies weakly from Li (r_{s} about 3.3) to Cs (r_{s} about 5.6), whereas the individual components of the work function vary quite a lot, as was seen in lecture 4, Table 4.1. It is this feature, the substantial internal cancellation of terms, which made the calculation a challenge in 1970.

Figure 3: Work functions in the jellium model (full squares, Lang and 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 4thorder polynomial fit to these points has been added.

Figure 4: A ‘quantum corral’ of 60 Fe atoms assembled and viewed on Cu(001) by STM at 4 K. The tip imaging parameters are a) V_{t} = + 10 mV and b)  10 mV, with current I = 1 nA (after Crommie et al. 1995).
Whether or not these effects can be explained in detail as yet (Fe and Cu are both transition metals with important dbands), these oscillations are present in free electron theory. To see how such effects 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 and Kohn goes roughly as follows, using figure 5 as a guide.
Figure 5: a) cross section of the free electron Fermi surface, radius k_{F}; b) the combination of traveling wave states ±k near a surface. See text for discussion.
Consider pairs of states, ordered by their kvector perpendicular to the surface, k and k. Their wavefunction is y ~ y_{k}(z)exp i(k_{x}x + k_{y}y), and when ±k are combined to vanish in the vacuum (outside the surface), y_{k}(z) ~ sin(kz g_{F}), where g_{F} is a phase factor, dependent on k_{F}, since the origin doesn’t have to be exactly at z = 0.
Draw a Fermi sphere, radius k_{F}, with the kaxis (perpendicular to the surface) as a unique axis, as in figure 5. Make a slice at k, dk thick; the density of states g(k) is just the area of this slice which is p (k_{F}^{2}  k^{2}). Now we can write
where the limits of integration are 0 and k_{F}, and with a bit of manipulation you should get the result
where the Onotation means ‘of order (2k_{F}z)^{3} ’. Here nbar is the electron density in the bulk; the symbols nbar and r^{} are used interchangeably. The point which is specific to 2D surfaces and interfaces is the dependence on (2k_{F}z)^{2}. For impurities or point defects, the result is O(2k_{F}z)^{3}, which is due to 3D geometry. For corrals on the surface with cylindrical geometry, we encounter various types of Bessel function, as with many other cylindrically symmetric problems. In scattering/ perturbation theory terms, the characteristic length, (2k_{F})^{1}, is due to scattering across the Fermi surface without change of energy. The same length occurs in the theory of superconductivity and charge density waves.
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, whereas Nb and W, which are otherwise similar, have variations of around 0.8 V.
The origin of this facespecific nature of the work function can be seen qualitatively by considering jellium again. First, we can see from figure 2a that the negative charge spills over into the vacuum, causing a dipole layer, whose dipole moment is directed into the metal. Now we use Gauss’ law and show that
where the sheets of charge, surface charge density s, are separated by a distance d. To get an idea of how big the potential change is, think of each atom on the surface (N m^{2}) having a charge of 1 electron separated by 0.1 nm (1 Ångström). With N = 2x10^{19} m^{2}, p = 1.6 x10^{29} Cm, and e_{0} = 8.854 x10^{12} Fm^{1}, we get DV = 36.14 V. This value is perhaps 25 times as large as most voltage (energy) differences between the vacuum level and the bottom of the valence band (which is also the conduction band in monovalent metals).
So a charge separation of < 0.5Å is needed to produce the desired effect. Is it a coincidence that this is the same order of magnitude as the Bohr radius, a_{0} = 0.0529 nm? Not really: the reasons for both effects, the spill over of electrons due to the need to reduce kinetic energy, are the same! This is, of course, a zero order argument: to get the numbers right we have to go back to exchange and correlation energies, and the details. However, models may contain rather arbitrary parameters. For example, the ‘corrugation factor’ introduced into the ‘structureless pseudopotential model’ (Perdew et al. 1990) sounds rather dubious, although it moves the model in the right direction (Perdew 1995). Brodie (1995) has proposed a model, building on the idea of corrugation, which is rather ‘too simple’; this model should be ignored since it is incapable of further elaboration.
While on this subject, we can note the unit to describe dipole moments, the Debye (D). This is 10^{18} esu.cm = 3.33 x 10^{30} Cm. Thus 1 electron charge x 1Å = 4.81 D. Adsorbed atoms change the work function considerably, but only alkalis give rise to dipole moments this large; for example Cs adsorbed on W(110) at low coverages has been calculated to have a dipole moment of at least 9D (see e.g. table 2.2. in Hölzl & Schulte 1979); in this case the single electron charge distribution would be shifted by about d = 0.2 nm. This simple picture is illustrated in figure 6a, and corresponds to (partial) ionization of the alkali, a model first introduced by Langmuir in 1932 and developed by Gurney in 1935. But we need to be careful about inclusion of the image charge, and the nature of bonding, which varies with coverage and is the subject of ongoing discussion (Diehl & McGrath 1997).
Figure 6: Origins of faceand adsorbatespecific work function: a) dipoles due to charge transfer from adsorbates; b) topview of a stepped surface showing smoothing of the charge distribution around the steps (after Gomer 1961 and Woodruff & Delchar 1986).
The same arguments about electron spillover tell us that stepped, or rough surfaces will have lower work function than smooth surfaces, due to dipoles associated with steps, pointing in the opposite direction to the dipole previously considered for the flat surface. A schematic (top view) of this situation, referred to sometimes as the Smoluchowski effect after a 1941 paper, is shown in figure 6b. Experiments on vicinal surfaces, close to low index terraces, do indeed show that the work function decreases linearly with step density, as shown in figure 7; this implies that there is a well defined dipole moment per ledge atom, around 0.3 D for steps parallel to [001] on W(110) and varying with step direction on Au and Pt(111) surfaces (KrahlUrban et al. 1977; Besocke et al. 1977; Wagner, 1979).
(a) 
(b) 
Figure 7a: Vicinals of W(110) in the [001] zone as a function of step density; Figure 7b: Vicinals of Au and Pt(111) with two different step directions (KrahlUrban et al. 1977; Besocke et al. 1977; Wagner, 1979).
Work function changes as a function of temperature can be used to define 2D solidgas phase changes via these same effects. An adatom has a dipole moment which depends both on its chemical nature and on its environment. In a solid ML island the dipole moment per atom is considerably smaller than in the 2D gas. This effect has been used to map out the gassolid phase boundary for Au/W(110) at high temperatures (Kolaczkiewicz & Bauer 1984).
This book section further shows that the jellium model also gives, though not so impressively, values for the surface energy of the same metals. The agreement is again excellent for the heavier alkali metals, but fails dramatically for small r_{s}. This arises from the need to include the discreteness of the positive charge distribution associated with the ions, a point which was recognized in Lang and Kohn’s original paper. With a suitable choice of pseudopotential, agreement is much improved. A study of the surface energies of metals is a suitable project to follow on from the same book section.
We are pleased to acknowledge the permission of specific authors in allowing us to use their published figures in this project. Ben acknowledges that his period in Sussex is made possible by a period of leave, and support, from the University of Botswana.