1906 Electronic structure and emission processes

where the O-notation means `of order (2kF z)23 '. Here nÅ is the electron density in the bulk; the symbols nÅ and r2 are used interchangeably. The point which is speci®c to 2D

surfaces and interfaces is the dependence on (2kFz)22. For impurities or point defects, the result is O(2kFz)23, which is due to 3D geometry. For corrals on the surface with cylindrical geometry, we encounter various types of Bessel function, for the same

reasons as in chapter 5. In scattering/perturbation theory terms, the characteristic

length, (2kF)21, 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;

these features can be explored further via problem 6.1.

It is interesting that the jellium model also gives, though not so impressively, values for the surface energy of the same metals as shown later in ®gure 6.10. The agreement

is again excellent for the heavier alkali metals, but fails dramatically for small rs. This arises from the need to include the discreteness of the positive charge distribution asso-

ciated with the ions, a point which was recognized in Lang & Kohn's original paper. With a suitable choice of pseudopotential, agreement is much improved (Perdew et al. 1990, Kiejna 1999).

6.1.2Beyond free electrons: work function, surface structure and energy

There have been many developments since Lang & Kohn to extend this approach, ®rst to s-p bonded metals and then to the complications of transition metals involving d- electrons, and in the case of the rare earths, f-electrons as well. The d-electrons give an angular character to the bonding, often resulting in structures which are not closepacked, e.g. b.c.c. (Fe, Mo, W, etc.) or complex structures like a-Mn. This is in contrast to s-p bonded metals which typically are either f.c.c. or h.c.p. There are many challenges left for models of metallic surfaces.

To start we need a few names of the methods, for example `nearly-free electron' method, pseudopotentials, orthogonalized plane waves (OPW), augmented plane waves (APW), Korringa±Kohn±Rostoker (KKR), tight-binding, etc. These longstanding methods are described by Ashcroft & Mermin (1976). For surfaces, an introductory account of electronic structure is given by Zangwill (1988), which contrasts with a highly detailed version from Desjonquères & Spanjaard (1996). Typically tightbinding (where interatomic overlap integrals are thought of as small) is taken as the opposite extreme to the nearly free electron model (where Fourier coeYcients of the lattice potential are thought of as small). However, this is more apparent than real, in that both pictures can work for arbitrarily large overlap integrals or lattice potentials; the only requirement is that the basis sets are complete for the problem being studied. This of course can lead to some semantic problems: methods which sound diVerent may not in fact be so diVerent; in particular, when additional eVects are included they 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; an early calculation which shows this for the lowest atomic number metal lithium is given in ®gure 6.6. So there are now two length scales in the problem which

6.1 The electron gas	191

Figure 6.6. Valence electron density at several x// points for Li(001) in a pseudopotential calculation (from Alldredge & Kleinman 1974, reproduced with permission, after Appelbaum & Hamann, 1976).

compete; surface states have oscillation periods with no simple relation to the lattice period in the z-direction.

Contrary to the free electron starting point, it has more recently proved fruitful to consider models based on wavefunctions relatively localized in real, rather than reciprocal, space, and to construct interatomic potential functions arising from atomic-like entities interacting with the electron gas in which the `atom' is embedded (Sutton 1994, Pettifor 1995, Sutton & BalluY 1995). The resulting methods are known as embedded atom models (EAM) or eVective medium theories (EMT); in these models the embedding energy DE is expressed in terms of the cohesive function Ec(n), as

DE5Ec(n)1DEc,

(6.5)

where the correction energy DEc diVers between the various schemes, but is relatively small.

The cohesive function Ec(n) is a function of the homogeneous electron gas density n in which the atom is embedded (Jacobsen et al. 1987, Jacobsen 1988, Nùrskov et al. 1993). The cohesive energy, and the component Ec(nÅ) at the optimum density nÅ is shown for the 3d transition metals in ®gure 6.7. A major eVect of these models is to

192 6 Electronic structure and emission processes

(eV/atom)gy	0	EMT
	4
	2	Modified EMT
		Moruzzi et al.
ener	6
Cohesive
	8
(a.u.)	4

s rRadius,

3

2

K Ca Sc Ti V Cr Mn Fe Co Ni Cu

Figure 6.7. Calculated cohesive energies and the equilibrium radius rs for the 3d transition series, comparing eVective medium theory (EMT) (open circles) with KKR methods (Morruzzi et al. 1978, closed circles). The modi®ed EMT (open squares) corresponds to EMT applied at the density given by the KKR method (Jacobsen et al. 1987, and Jacobsen 1988; redrawn with permission).

show clearly that metallic binding is strongly non-linear with coordination number. The ®rst `bonds' to form are strong, and get progressively weaker as extra metal atoms are added to the ®rst coordination shell. Some of these eVects were exhibited by the experimental examples described in section 5.4. There are many subtleties in the 3d series resulting from magnetism; here the overall cohesion peaks before and after the middle of the transition series, unlike the 4d and 5d series, where cohesion from the d- bands peaks in the middle. Note that this particular calculation does not include spincorrelation eVects, but some of these are discussed in relation to magnetism in section 6.3.

Many metal surface relaxations and reconstructions are due to competing electronic and vibrational eVects of a quite complex kind. For example, reconstructions of transition metals are often subtle, such as the W(001)231 and the `almost 231' incommensurate Mo(001) structures at low temperatures mentioned in section 1.4.3. These structures are driven by (angular) bonding instabilities at low temperature and by anharmonic lattice dynamics at high temperature (Ingles®eld 1985, Estrup 1994, Titmus et al. 1996). F.c.c. noble metal surfaces can be strongly aVected by their d-elec- trons, interacting with the ions and the other electrons. Although Ag(111)(131) is unreconstructed, Au(111) has a uniaxially compressed herringbone (roughly 2331)

6.1 The electron gas	193

reconstruction, Au(001) has a quasi-hexagonal surface layer giving a (roughly 2035) diVraction pattern (Van Hove et al. 1981, Barth et al. 1990), and Pt(111) has a 737 reconstruction which can be removed by depositing Pt adatoms (Needs et al. 1991, Bott et al. 1993). This last case shows that the surface structure and lattice parameter of a metal can be a function of the supersaturation Dm of its own vapor, and that adatoms and surface reconstructions can change the surface stress. This possibility is also well known from adsorption studies of rare gases on graphite, as discussed in section 4.4.

An increasing number of theorists are suYciently practical and public-spirited that they collaborate closely with experimentalists, and make their computer codes available to others for work on speci®c 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 Schrödinger equation that they can think of, but that this doesn't necessarily help one understand trends in behavior, or help one make predictions. Pettifor (1995), 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 theoretical models than we would be able to otherwise.

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; O(N) methods are in! For example, because the interactions between atoms and the electron gas are parameterized initially, EMT calculations are fast enough that they can be used to simulate dynamic processes such as adsorption, nucleation or melting on metal surfaces; here an approximate electronic structure calculation is being done for each set of positions of the nuclei, i.e. at each time step (Jacobsen et al. 1987; Stolze 1994, 1997). This requires computer speeds that would have been inconceivable just a few years ago. It is now feasible to download EMT programs from a website in Denmark (see Appendix D) in order to run them for a class project in Arizona! There are real possibilities for experiment±theory collaborations here which were impractical just a few years ago.

6.1.3Values of the work function

There are several methods of measuring the work function, as described by WoodruV & Delchar (1986, 1994), by Swanson & Davis (1985) and by Hölzl & Schulte (1979) amongst others. The work function varies with the surface face exposed, as shown for several elemental solids in table 6.2. Note that for b.c.c. metals, the surfaces decrease in roughness in the order (111), (100), (110) presented, whereas for f.c.c. the same order corresponds to an increase in roughness. These variations are responsible for several interesting eVects, as described here and in the next section.

A polycrystalline material, with diVerent faces exposed, gives rise to ®elds outside the surface, referred to as patch ®elds. Such ®elds are very important for low energy electrons or ions in vacuum, and can thereby in¯uence measurement accuracy in surface experiments. Molybdenum is often used for such critical parts of UHV apparatus,

1946 Electronic structure and emission processes

Table 6.2. Experimental work functions for metals assembled by Michaelson (1977), compared with calculations by Perdew, Tran & Smith (1990; PTS), Skriver & Rosengaard (1992; SR) and Methfessel, Hennig & ScheZer (1992; MHS), plus others as indicated in the last column

					Model
Metal/	Face	Experiment*	Model	Model	(MHS)
structure	{hkl}	(eV)	(PTS)	(SR)	1 others*
Li	111		2.90
b.c.c.	100	2.9 (poly)	2.92	3.15	3.03b
	110		3.09	3.33	3.27b
Na	111		2.54
b.c.c.	100	2.75 (poly)	2.58	2.76	2.66b
	110		2.75	2.94	2.88b
K	111		2.17
b.c.c.	100	2.30 (poly)	2.21	2.34	2.27b
	110		2.37	2.38	2.44b
Cs	111		1.97
b.c.c.	100	2.14 (poly)	2.01	2.03	2.04b
	110		2.17	2.09	2.19b
Al	110	4.06	3.81
f.c.c.	100	4.4160.03a	3.62		4.50a
	111	4.24	3.72	4.54	4.09b
Cu	110	4.48		4.48
f.c.c.	100	4.59		5.26
	111	4.98		5.30
Ag	110	4.52		4.40	4.23
f.c.c.	100	4.64		5.02	4.43
	111	4.74		5.01	4.67
Au	110	5.37		5.40
f.c.c.	100	5.47		6.16
	111	5.31		6.01
Nb	111	4.36
b.c.c.	100	4.02			3.68
	110	4.87		4.80	4.66
Mo	111	4.55
b.c.c.	100	4.53			4.49
	110	4.95		5.34	4.98
W	111	4.47
b.c.c.	100	4.63
	110	5.25		5.62

Note: *Error bars and other calculations by: (a) Ingles®eld & Benesh (1988); (b) Perdew (1995).

6.1 The electron gas	195

because the work function doesn't vary more than 0.4 V between the low index faces (table 6.2), whereas Nb and W, which are otherwise similar, have variations of around 0.8 V.

The origin of this face-speci®c nature of the work function can be seen qualitatively by considering jellium again. First, we can see from ®gures 6.1 and 6.2 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

DV(volts)5sd/«05pN/«0,

(6.6)

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 (Nm22) having a charge of 1 electron separated by 0.1 nm (1 ångström). With N523 1019 m22, p51.63 10229 Cm, and «058.8543 10212 Fm21, we get DV536.14 V. This value is perhaps 2±5 times as large as most voltage (energy) diVerences 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 eVect. Is it a coincidence that this is the same order of magnitude as the Bohr radius, a050.0529 nm? Not really: the reasons for both eVects, 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 `too simple' in Einstein's sense; 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 10218 esu´cm53.33310230 Cm. Thus 1 electron charge31Å54.81 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 and Schulte 1979); in this case the single electron charge distribution would be shifted by about d50.2 nm. This simple picture is illustrated in ®gure 6.8(a), and corresponds to (partial) ionization of the alkali, a model ®rst 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).

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 ¯at surface. A schematic (top view) of this situation, referred to sometimes as the Smoluchowski eVect after a 1941 paper, is shown in ®gure 6.8(b). Experiments on vicinal surfaces, close to low index terraces, do indeed show that the work function