NAN/PHY/MSE 546 (Venables) Sect 1.5

Notes for NAN/PHY/MSE 546 Sect 1.5 (Venables)

© Arizona Board of Regents for Arizona State University and John A. Venables

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Lecture notes by John A. Venables. Revised for Spring '03; Latest version 25 August 2012, reformatted. I am still looking for the unicode versions of Symbol "f", as in the diagram for the work function. For the moment I am using φ and , so I hope there is no confusion. The .doc version is clear in any case.

1.5 Introduction to Surface Electronics

Refs: Prutton, Chap 4; Zangwill, Chap. 4; Lüth, Chap 6; More specialist review articles, such as J.E. Inglesfield, in D.A. King and D.P. Woodruff (eds) The Chemical Physics of Solid Surfaces and Heterogeneous Catalysis, vol 1, especially sects 1, 2.1 and 2.2. There are many such articles, and various specialist books. Here we are concerned only to define and understand a few terms which will be used in a general context; the discussion is at a similar level of detail to my book, section 1.5. A detailed study of electronic effects can be done later.

The terms which we will need include the following. I will draw the corresponding diagrams on the board and discuss them:

Work Function

This is the energy, typically a few eV, required to move an electron from the Fermi Level, EF, to the vacuum level, E0. The work function depends on the crystal face {hkl} and rough surfaces typically have lower work function, φ as discussed later in section A1.

Electron Affinity and Ionisation Potential

Both of these would be the same for a metal, and equal to φ, but for a semiconductor or insulator, they are different. The electron affinity is the difference between the vacuum level E0, and the bottom of the Conduction Band EC. The ionisation potential is E0 - EV, where EV is the top of the valence band. These terms are not specific 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.

Surface States and related ideas

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

ψ ~ u(r) exp (-i kz|z|) exp (i k// r),

where, for a state in the band gap, kz is complex, leading to decay away from the surface on both sides. 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 continuosly into a bulk state. A surface plasmon is a collective excitation located at the surface, with frequency typically ωp/√2.

Surface Brillouin Zone

A surface state takes the form of a Bloch wave in the 2-dimensions of the surface, in which there can be energy dispersion as a function of the k// (parallel) vector. For electrons crossing the surface barrier, k// is conserved, kz (perpendicular) is not. The k// conservation is to within a 2D reciprocal lattice vector, i.e. ±G//. This is the theoretical basis of (electron and other) diffraction from surfaces.

Band 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 you can convince yourself that a p-type semiconductor has bands which are bent downwards as you approach the surface. This leads to a reduction in the electron affinity. Some materials (eg Cs/p-type GaAs) can even be activated to negative electron affinity, and such materials form a potent source of electrons, which can also be spin-polarised as a result of the band structure.

The Image Force

You will recall from elementary electrostatics that a charge outside a conducting plane has a field on it equivalent to that produced by a ficticious ‘image charge’. The corresponding potential felt by the electron, V(z) = -e/4z. For a dielectric, with permitivity ε, there is also a (reduced) potential

V(z) = -(e/4z) (ε-1)/(ε+1).

It is often useful to think of metals as the limit ε→∞, and vacuum as ε→ 1. The diagram showing the lines of electric field from an incident electron in the dielectric case is on the cover of Lüth’s book, and is described in section 4.6.2 (3rd edition, page 165).(Note added 13 May '02) In this section, I have been too sloppy in the formulae and have used old-fashioned c.g.s units without thinking. In MKS (S.I.) units, these formulae should be multiplied by the constant K = 1/(4πε0). Thanks to Richard Forbes and Michael Isaacson for spotting this, and to Yong Jiang for working through the related problem 1.6 in Spring 2002.

Screening

The above description emphasises the importance of screening, in general, and also in connection with surfaces. We can also notice the very different length scales involved in screening, from atomic dimensions in metals, (2kF)-1, increasing through narrow and wide band gap semiconductors to insulators, and vacuum; there is no screening (at our type of energies!), unless many ions and electrons are present ( i.e. a plasma). In general, nature tries very hard to remove long range (electric and magnetic) fields, which contribute unwanted macroscopic energies. We will come back to this point, which runs throughout the physics of defects; in this sense, the surface is simply another defect with a planar geometry.


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