196 6 Electronic structure and emission processes

	d	+ve
		-ve

charges

(a)									(b)
					image charges

Figure 6.8. Origins of faceand adsorbate-speci®c work function: (a) dipoles due to charge transfer from adsorbates; (b) top-view of a stepped surface showing smoothing of the charge distribution around the steps (after Gomer 1961, and WoodruV & Delchar 1986, redrawn with permission).

decreases linearly with step density, as shown in ®gure 6.9; this implies that there is a well de®ned 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 (Krahl-Urban et al. 1977; Besocke et al. 1977; Wagner 1979).

Fascinatingly, work function changes as a function of temperature can be used to de®ne 2D solid±gas phase changes via these same eVects. 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 eVect has been used to map out the gas±solid phase boundary for Au/W(110) at high temperatures (Kolaczkiewicz & Bauer 1984). Phase changes in adsorbed layers are discussed in more detail in chapter 4.

6.1.4Values of the surface energy

While the work function is a sensitive test of electronic structure models, particularly exchange and correlation, surface energies are sensitive tests of our understanding of cohesion. Surface energies clearly increase, in general, with sublimation energies, and this has led to many studies trying to embody such relationships into universal potential curves, or into other semi-empirical models (Rose et al. 1983, deBoer et al. 1988). But the eVective medium and density functional models have proven to be more durable, and we may well now have arrived at the point where these models are more accurate than the experiments, which were almost all done a long time ago on polycrystalline samples, sometimes under uncertain vacuum conditions. An example of a microscope based sublimation experiment for Ag, which has stood the test of time, and which agrees with recent calculations within error, is that by Sambles et al. (1970). Some experimental and theoretical values for a range of metals are given in table 6.3.

As in the case of the work function, the starting point for models of alkali and other s-p bonded metals has been the jellium model. As was shown in the original Lang &

Kohn papers, jellium has an instability at small rs values, which is due to the neglect of the ion cores. This topic has been pursued by Perdew et al. (1990), Perdew (1995) and

Kiejna (1999), who have been interested in exploring the simplest feasible pseudopotential models for such metals, and in particular obtaining trends in calculations as a

6.1 The electron gas	197

Figure 6.9. Work function of stepped (vicinal) surfaces: (a) vicinals of W(110) in the [001] zone as a function of step density; (b) vicinals of Au and Pt(111) with two diVerent step directions (Krahl-Urban et al. 1977; Besocke et al. 1977; Wagner 1979, reproduced with permission).

198 6 Electronic structure and emission processes

	1.2
	Al
	1.0	Zn
	Ga
)	0.8	Mg
±2
(J.m		Cd
energy		In
	0.6	Hg
	Pb		Li
Surface		Ca	Sr
	0.4			Ba
				Na
	0.2			K	Rb
						Cs
	0.0
	2	3		4	5
		radius,		rs (a.u.)

Figure 6.10. Surface energies of alkali and other s-p bonded metals, showing experimental values on polycrystalline materials and liquid metals (full squares from Tyson & Miller 1977 and Vitos et al. 1998, with open squares from deBoer et al. 1988 where the results diVer signi®cantly), compared with the jellium (dashed curve, sixth order polynomial ®t) and ¯at structureless pseudopotential models (full curve, fourth order polynomial ®t after Perdew et al. 1990) as a function of the radius rs. The jellium model has an instability at small rs, which is pushed to smaller values in the stabilized model.

function of rs as illustrated in ®gure 6.10. The model illustrated is termed the structureless pseudopotential model, and has been developed in various varieties; the one illustrated here is `¯at' in the sense that no attempt is made to take account of the actual ionic positions on diVerent {hkl} faces. There are many other calculations in the literature, especially aiming to take account of the complexity of d-band metals, some of which are cited in table 6.3; several of these calculations can be accessed from the data base for 60 elements compiled by Vitos et al. (1998).

In section 1.2.3, we noted that pair bond models overestimated the anisotropy of surface energy in comparison with the classic experiments of Heyraud & Métois (1983) on Pb, which were shown in ®gures 1.7 and 1.8. The modern calculations shown in table 6.3 are the only ones getting the anisotropy of surface energy low enough to be even close to experiments on small metal particles. However, it is ironic that the full charge density (FCD) calculation, carried out with the GGA approximation by Vitos et al. (1998) still gives too high an anisotropy, especially for Pb, which is almost the only case (plus In and Sn, see Pavlovska et al. 1989, 1994) for which there

6.1 The electron gas	199

Table 6.3. Experimental surface free energies for metals assembled by Eustathopoulos et al. (1973), Tyson & Miller (1977), Mezey & Giber (1982) and deBoer et al. (1988), compared with calculations by Perdew, Tran & Smith (1990; PTS), Skriver & Rosengaard (1992; SR), Methfessel, Hennig & ScheZer (1992; MHS) and Vitos, Ruban, Skriver & Kollár (1998; VRSK)

	Experiment*				Model
Metal/	(J/m2), at T	Face	Model	Model	(MHS)	Model
Structure	and at 0 K	{hkl}	(PTS)	(SR)	1 others*	(VRSK)
Li		111	0.433
b.c.c.	0.525	100	0.371	0.436	0.412b
		110	0.326	0.458	0.362b
Na		111	0.252
b.c.c.	0.260	100	0.216	0.236	0.237b
		110	0.190	0.307	0.208b
K		111	0.134
b.c.c.	0.130	100	0.115	0.129	0.126b
		110	0.134	0.116	0.111b
Cs		111	0.092
b.c.c.	0.095	100	0.079	0.092	0.080b
		110	0.069	0.072	0.070b
Al	1.14	110	1.103			1.271
f.c.c.	175°C	100	0.977			1.347
	1.14±1.16	111	0.921	1.27	1.096b	1.199
Cu	1.60	110		2.31		2.237
f.c.c.	900±1070°C	100		2.09		2.166
	1.78±1.83	111		1.96		1.952
Ag	1.20 6 0.06a	110		1.29	1.26	1.238
f.c.c.	800±830°C	100		1.20	1.21	1.200
	1.25	111		1.12	1.21	1.172
Au	1.40 6 0.05c	110		1.79		1.700
f.c.c.	950±1000°C	100		1.71		1.627
	1.50	111		1.61		1.283
Nb	2.30	111				3.045
b.c.c.	1900°C	100			2.36	2.858
	2.67±2.70	110		1.64	2.86	2.685
Mo	2.00	111				3.740
b.c.c.	1800°C	100			3.14	3.837
	2.95±3.00	110		3.18	3.52	3.454
W	2.80	111				4.452
b.c.c.	1700°C	100				4.635
	3.25±3.68	110		3.84		4.005

Note: *Error bars and other calculations from (a) Sambles et al. (1970); (b) Perdew (1995);

(c) Heyraud & Métois (1980).