Lecture notes by John A. Venables, revised for Spring 2005.

Refs: Review and specialist articles and books referred to in the text (see list in the previous section). This section has been expanded slightly in my book, chapter 3.4.

The ratio I_{A}/I_{p} is a product of terms describing the production
and detection of the Auger electrons. The *Auger yield* Y is the number of Auger electrons
emitted into the total solid angle (Ω = 4π sterad).
It is therefore not dependent on the details of the analyser.
The *detection efficiency* D of the analyser can be written as (T.ε),
where T is a function f(Ω_{a}/4π), Ω_{a} being the solid angle
collected by the analyser, and ε is f(ΔE/E), most simply = (ΔE/E). Thus

Here we have Y expressed as the cross-section for the initial
ionization event (σ), the Auger efficiency (γ), discussed in outline in section 3.3,
and the factor R. The secθ_{o} term describes the extra ionization path length
caused by having the primary beam at an angle to the sample normal. Finally N_{e} is
the effective number of atoms/unit area contributing to the (particular) Auger process.

What we actually want to know is: given a measured signal I_{A}, how many A-atoms are there
on the surface? Typically there is not a unique answer to such a simple question, because the signal
depends not only on the number of atoms but also on their distribution in depth (diagram 114). There
are two cases which can be solved uniquely, which are instructive in showing how such analyses work.
The first is when all the atoms are in the surface layer: then N_{e} = N_{1}, and if
one knew all the other terms in the equation, we could determine N_{1}. The second case is
when the atoms are uniformly distributed in depth: in this case we can show that N_{e} =
N_{m}, where N_{m} is the bulk (3D) density of A-atoms. The proof of the second case
is as follows. We work out

The angular double integral is just the cosine electron emission distribution, integrated over the
solid angle of the analyzer; so f = T, N_{e} = N_{m}, as required.

A detailed study of the Auger signals from bulk elements has been performed by D.R. Batchelor
*et al.*, Surface and Interface Analysis **14** (1989) 700-716, from which diagrams
115 and 116 are taken. These studies show that the dependence of the Auger peak height on primary
beam energy E_{o} explores the variation of (σR); the variation
with θ_{o} is determined by
(Rsecθ_{o}.T); and the dependence on atomic number Z is
influenced by σ, γ, R, N_{m} and
ε. To make these comparisons we need to take a particular ionization
cross-section, and also to calculate the Auger backscattering factor R = 1 + r, where r is defined
as

where the normal electron backscattering factor η is given by

This means that the Auger backscattering factor can be written as R = 1 + βη, where the factor β is typically greater than 1, and depends on the energy and angular distribution of backscattered electrons, and the Auger energy in relation to the beam energy via the relevant cross sections. The detailed study of the basic quantification equation thus leads to some interesting physics, and maybe eventually to a determination of one or more parameters of the experiment, but it does not lead directly to easy analysis of samples. For this we have to keep many of the experimental parameters fixed, and use standard samples for comparison.

the last bracket is what we want to know, and the previous term is a matrix dependent factor. Without detailed calculation it is not obvious how such terms behave, but they can sometimes vary slowly. For example, P.H. Holloway (University of Florida) and his group have studied several binary alloy systems, and shown that the matrix dependent factors often vary linearly with composition. If one is stuck, one could establish standards closer to the composition of interest. This might be seen as a last resort: it is clear that the smaller extrapolation you have to perform, the more accurate the result is likely to be.

Many authors have developed programs for studying such effects, and various national standards organizations (NIST, NPL, etc) have got involved. The author most concerned in the USA is C.J. Powell of NIST, who has written extensively on this topic; his UK counterpart is M.P. Seah, who works for NPL. It is clearly no accident that they have concerned themselves with these aspects of quantitative analysis, and indeed run conferences with the title 'Quantitative Surface Analysis', or journals such as 'Surface and Interface Analysis' at/in which such matters are discussed in great detail. There are also major consulting businesses based on such analyses, since the services and expertise are often very expensive to maintain 'in-house'; the best known may be C.A. Evans and Associates. I do not propose to discuss these topics further here, but the flavor of this work can be obtained from the specialist books, especially G.C. Smith's; he too worked for NPL, and gives a list of significant papers by the above authors and others. Some of these resources are featured on my Surface Analysis page.

A typical 'surface science' application is to layer by layer growth. This is a case which you can work
out for yourself; the experimental ratio is (I_{A}/I_{p}) for a multilayer divided by
(I_{A}/I_{p}) for a ML. Here we assume that there are N_{1} atoms in the first
layer, N_{2} in the second and so on, and their spacing is d. Then we can work out the signals
at coverage between n and n+1 ML from both the layers and the substrate, as sums which take into
account the attenuation. For example, if n = 1, and we neglect attenuation within the first layer

where the first (second) term in brackets correspond to the proportion of the first layer which is
uncovered (covered) by the second layer, and so on. This relation leads to a series of straight lines
in curves for layer growth, often plotted as a function of deposition time (diagram 111), from which
the *imfp* λ can be deduced in favorable cases.

The simplest case, where there are the same numbers of atoms in each completed ML, can be worked out
explicitly. In that case, the slope of the second ML curve ratioed to that of the first ML gives
exp(-d/λcosθ_{e}), from which d/λ can be extracted if the effective analyser angle
θ_{e} is known. This effective angle is given by

with the integrals taken over the analyzer acceptance. For detailed studies, it is advisable to construct computer programs which take the analyzer geometry into account, and then perform the angular integration numerically. The purists among you should note that what is determined experimentally is known as the attenuation length; the discussion of how elastic (wide angle) scattering influences the relation of experiment to the imfp is ongoing (see Smith, section 5.3, pages 58-61 and refs quoted).

An example of a layer growth analysis is shown in diagram 112, which shows both the Auger spectra for a series of Ag deposits on Si(100) and the corresponding Auger curves as a function of coverage at both room (RT) and elevated temperature. This is a case where the growth follows the ‘layer plus island’, or Stranski-Krastanov mode. From the Auger curves we see that at RT, the layer growth is followed for approximately 2ML, but then they diverge, indicating roughening or islanding. The high T behavior is much more extreme, with a first layer which is less than 0.5 ML thick, and islands grown on top of, and in competion with, this dilute layer. The Ag/Si(111) and Ag/Ge(111) systems have also been studied with the root-3 reconstructed layer at high temperatures having a thickness of around 1 ML. The exact coverage of such layers has been surprisingly controversial, especially when the layers are only slightly more stable than the islands.

There are now many examples, from this and other growth systems, which show characteristic Auger curves
associated with layer and island growth. The purpose of a detailed layer growth analysis is often to show
that the system does *not* follow the layer growth mode. It is only by detailed quantitative analysis,
with for example known mean free paths, that one can make such definitive statements from Auger curves
on their own. An example in which I was involved is J.A. Venables *et al.*, J. Phys D **29**
(1996) 240-245, with corrections in J. Phys D **30** (1997) 3163-5, and more examples in R. Persaud
*et al.*, Surface Sci. **401** (1998) 12-21. In this case we were using the layer growth analysis
to investigate surface segregation in Si/Ge/Si(100) and inter-diffusion in Fe/Ag/Fe(110) hetero-structures.

You might note that many authors have used *deposition time* as the dose variable, and then simply state
that the break point corresponds to 1 ML coverage; this is poor logic, since the coverage is the
*independent* variable and the Auger intensity is the *dependent* variable. Used in the way
described here, the Auger curve is not calibrated absolutely; it needs to be calibrated indepentently, e.g.
via a calibrated quartz oscillator or Rutherford Backscattering Spectroscopy (RBS), as discussed,
for example, by Feldman and Mayer.

**A possible mini-project on Surface Analysis:** Write computer
programs for the Auger intensity as a function of coverage for element A growing
on element B in the following cases. The effective mean free paths are
λ_{AA} for Auger line A at Energy E_{A}
in element A, λ_{AB} for Auger line A at
Energy E_{A} in element B, with λ_{BA}
and λ_{BB} defined analogously.

(a) "Almost" layer by layer growth, neglecting backscattering effects, where all
layers have the same atomic density, N_{0}, except that the first layer has
a different density N_{1} that is < N_{0}.

(b) Stranski-Krastanov growth, where islands of atomic density N_{0} and
height h grow on top of the first layer with density N_{1} that is < N_{0}.
What can you say about backscattering effects if the backscattering coefficient of
element B is greater than that of element A?