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*

**Click to download this document in Microsoft
Word Format**

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*, E_{F},
to the vacuum level, E_{0}. 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 E_{0}, and the bottom of the *Conduction Band*
E_{C}. The **ionisation potential** is E_{0} - E_{V},
where E_{V} 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 k**_{z}|z|)
exp (i k_{//} r),
where, for a state in the band gap, k_{z} 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 ω_{p}/√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, k_{z}
(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/(4πε_{0}). 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,
(2k_{F})^{-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.

Continue to section 2.1
Return to Lecture list