301 Introduction to surface processes

again is a specialist topic, combining surface structure with surface electronics, that we consider in chapter 7.

1.4.8Ionic crystal structures, such as NaCl, CaF2, MgO or alumina

Here we have to consider the movement of the two diVerent charged ions, likely to be in opposite directions, and the resulting charge balance in the presence of the dielectric substrate. However, this `rumpling' is often found to be remarkably small, typically a few percent of the interplanar spacing; a ®rst point for a search of what is known experimentally and theoretically is the book by Henrich & Cox (1996). A recent development is to combine structural experiments (e.g. LEED) on ultra-thin ®lms grown on conducting substrates, to avoid problems of charging, with ab initio calculation. Some of these methods and results can be found in the atlas of Watson et al. (1996), review chapters in King & WoodruV (1997) and a 1999 conference proceedings on The Surface Science of Metal Oxides published as Faraday Disc. Chem. Soc. 114. Grazing incidence X-ray scattering is also helping to determine structures (Renaud 1998). A notable

exception to the general rule is alumina (A12O3), where the surface oxygen ion relaxations have been calculated to reach around 50% of the layer spacing on the hexagonal

(0001) face (Verdozzi et al. 1999). But are we getting ahead of ourselves: you can see how soon we need to read the original literature, but we do need some more background ®rst!

1.5Introduction to surface electronics

Here we are concerned only to de®ne and understand a few terms which will be used in a general context. The terms which we will need include the following.

1.5.1Work function, f

The work function is the energy, typically a few electronvolts, required to move an elec-

tron from the Fermi Level, EF, to the vacuum level, E0, as shown in ®gure 1.21(a). The work function depends on the crystal face {hkl} and rough surfaces typically have lower f, as discussed later in section 6.1.

1.5.2Electron af®nity, x, and ionization potential F

Both of these would be the same for a metal, and equal to f. But for a semiconductor or insulator, they are diVerent. The electron aYnity x is the diVerence between the

vacuum level E0, and the bottom of the conduction band EC, as shown in ®gure 1.21(b). The ionization potential F is E0 2 EV, where EV is the top of the valence band. These terms are not speci®c to surfaces: they are also used for atoms and molecules generally,

as the energy level which (a) the next electron goes into, and (b) the last electron comes from.

1.5 Introduction to surface electronics			31
E0		χ	E 0
φ			EC

Φ

EF

EV

(a)

(b)

Figure 1.21. Schematic diagrams of (a) the work function, f; (b) the electron aYnity, x and ionization potential F, both in relation to the vacuum level E0, the Fermi energy EF, and conduction and valence band edges EC and EV.

Figure 1.22. Schematic diagrams of: (a) a surface state de®ned by wave vector k//5kx1ky, and k'5kz; (b) the surface Brillouin zone and 2D reciprocal lattice vector G// for the Î33Î3R30° structure, plotted in the same orientation as the real (xenon) lattice of ®gure 1.16.

1.5.3Surface states and related ideas

A surface state is a state localized at the surface, which decays exponentially into the bulk, but which may travel along the surface. The wave function is typically of the form

c < u(r)exp (2 i k' |z|) exp (i k// r),

(1.18)

where, for a state in the band gap, k' is complex, decaying away from the surface on both sides, as shown in ®gure 1.22(a). Such a state is called a resonance if it overlaps with a bulk band, as then it may have an increased amplitude at the surface, but evolves continuously into a bulk state. A surface plasmon is a collective excitation located at the surface, with frequency typically vp/Î2, where vp is the frequency of a bulk plasmon.

32	1 Introduction to surface processes
	χ	E 0
	EC
	EV	EF	e+	± e
				z = 0
		(a)		(b)

Figure 1.23. Schematic diagrams of: (a) band bending due to a surface state on a p-type semiconductor; (b) the E-®eld between an electron at position z and a metal surface is the same as that produced by a positive image charge at 2 z.

1.5.4Surface Brillouin zone

A surface state takes the form of a Bloch wave in the two dimensions of the surface, in

which there can be energy dispersion as a function of the k// vector. For electrons crossing the surface barrier, k// is conserved, k' is not. The k// conservation is to within a 2D reciprocal lattice vector, i.e. 6 G//. This is the theoretical basis of (electron and other) diVraction from surfaces. For electrons there are two states contained in the surface Brillouin zone, which is illustrated for the hexagonal lattice of the Î33Î3R30° struc-

ture in ®gure 1.22(b).

1.5.5Band bending, due to surface states

In a semiconductor, the bands can be bent near the surface due to surface states. Under zero bias, the Fermi level has to be `level', and this level typically goes through the surface states which lie in the band gap. Thus one can convince oneself that a p-type semiconductor has bands that are bent downwards as the surface is approached from inside the material, as shown in ®gure 1.23(a). This leads to a reduction in the electron aYnity. Some materials (e.g. Cs/p-type GaAs) can even be activated to negative electron aYnity, and such NEA materials form a potent source of electrons, which can also be spin-polarized as a result of the band structure.

1.5.6The image force

You will recall from elementary electrostatics that a charge outside a conducting plane has a ®eld on it equivalent to that produced by a ®ctitious image charge, as sketched in ®gure 1.23(b). The corresponding potential felt by the electron, V(z)52 e/4z. For a die-

lectric, with permitivity «, there is also a (reduced) potential V(z)52(e/4z) («21)/ («1 1). It is often useful to think of metals as the limit «→`, and vacuum as « →1. Typical semiconductors have «,10, with «511.7 for Si and 16 for Ge; so semiconductors and

metals are fairly similar as far as dielectric response is concerned, even though they are not at all similar in respect of electrical conductivity.