PHY 598: Surfaces and Thin Films (Venables)
Second Assignment, Spring 2007
Problems set by John A. Venables. Latest version of this
document 10 February 2007.
The second assignments are due to be completed just after Spring break,
but I will set individual deadlines before then by email. I expect
one of the following problems to be done by the agreed deadline, and an
agreed project to be well underway. You have also done a
first assignment on section 1.
Make sure that you contact me in time if you need some help- you are not
expected to be able to do these assignments without some assistance.
Problem 2.1: Design of Vacuum Systems for specific purposes
Use your knowledge of (and the notes and appendices on) conductances of
standard size tubes, and the characteristics of vacuum pumps, to suggest
(and justify semi-quantitatively) design choices in the following situations:
- a) pumping an approximately spherical chamber of diameter 0.5m.
The chamber is to be let up to air infrequently, and we want to
acheive as good a pressure as possible (< 10-10 mbar).
- b) pumping a cylindrical chamber of length about 10m and diameter
0.1m. The chamber is to be periodically flooded with rare gases up
to about 10-3 mbar pressure, and the important point is to
be able to achieve pressures below 10-8 mbar quickly and
economically.
- c) pumping a state of the art particle accelerator from sections
of pipe of length about 50m and diameter 0.1m, with a total length
in excess of 50 km (kilometers) at a pressure of < 10-11 mbar.
Problem 2.2: Design of a Knudsen source for depositing
elemental metal films
A Knudsen source is an evaporation furnace which relies on the
establishment of the vapor pressure above a solid or liquid source
material. A small hole in the furnace above the source material, plus
collimating holes in front of the source allow a beam of the source
material to be directed at the sample. Use your knowledge of vapor
pressures and kinetic theory to design a source which will deposit one
monolayer per minute on a sample held 0.15 m away from the exit of the
source, will be uniform on the sample within 1% for the central 0.01 m
diameter, and will not deposit any material on the sample outside a
radius of 0.02 m. Do this in stages, with discussion, as follows:
- a) Consider the formula R = nv/4 for the number of atoms hitting
unit area per second of an enclosure, and how this formula applies to
a Knudsen source. Derive the formula by considering the relevant
integrations over angles and the Maxwell-Boltzmann velocity distribution.
- b) Consider the geometry of the design, and the constraints on the
uniformity and area of the deposit. Show that this will limit the size
of the hole in the furnace, and suggest a suitable size for holes in
both the furnace and the collimator.
- c) Choose an elemental metal of interest to you, and find out the
formula for the vapor pressure as a function of furnace temperature.
Using the relationship between density n and pressure p for this
material, coupled with your design from part b), work out the
temperature at which the source will have to operate to satisfy the
deposition rate requirement, explaining your assumptions.
- d) If you actually want to design a real source for this material,
consider carefully the materials from which the source can be made,
whether you should be using a Knudsen or some other type of source,
and how to power the furnace to achieve sufficient temperature
uniformity, etc. Or persuade your advisor to buy one....
Problem 2.3: Some questions on surface preparation and
related techniques
Questions about surface preparation are always very specific to the
materials concerned, but here are a few which may be relevant and
which spring from the text of this section.
- a) Why should one either cool the sample slowly through a surface
phase transition (e.g. as in the case of Si(111)), or not anneal the
sample above a bulk phase transition (e.g. in preparing b.c.c Fe
surfaces)?
- b) What is the main reason why Si(111) produces a 2x1
reconstruction after cleavage, when the equilibrium surface structure
is the 7x7?
- c) Device engineers always grow a ‘buffer layer’ on Si(100)
before attempting to grow a device, e.g. by molecular beam epitaxy.
Why is this precaution taken, and how does it improve the quality
of the devices grown on such surfaces?
- d) Mass spectrometry shows a range of mass numbers (M/e ratios)
for the contents of the vacuum system, but they don’t seem to be
simply related to the molecules, e.g. O2, N2,
CO, H2O, CO2 which are
present. What range of processes are responsible for this discrepancy?
- e) GaAs often evaporates to leave small liquid Ga droplets on the
surface. Why does this happen, and how can it be prevented?
Problem 3.1: Some Questions on Surface Techniques
Give a short description of the following points in relation to surface
techniques, including some examples:
- a) Explain why we say that we have conservation of k//, but not
of k^, in surface scattering experiments.
- b) Explain why surface X-ray Diffraction can be understood quantitatively
in terms of ‘kinematic’ scattering, whereas the various forms of Electron
Diffraction require a ‘dynamic’ theory.
- c) Explain why the lineshape in UPS is said to reflect the ‘valence band
density of states’ whereas the AES lineshape may depend on a ‘self-convolution
of the VB DOS’.
- d) Explain the experimental setup, and energy resolution, needed to observe
surface phonons. Comment on the relative energy resolution required for
inelastic photon (Raman), electron (HREELS) and Helium atom scattering.
Problem 3.2: The role of inelastic scattering in LEED
A quasi- kinematic model of LEED is possible based on the following
assumptions. The inner potential of the crystal, is V0 ~ 10 volts,
which increases the wavevector in the crystal over that in free space and
refracts the beam at the surface. The attenuation of the incident beam
amplitude (and the back-diffracted beams) is exponential with a short mean
free path l, which is inversely proportional to the
imaginary potential V0i ~ 3-5 volts. A single backscattering event
happens at a given atom at depth z, and has scattering factor f (or equivalently
t, the t-matrix) which is function of the beam energy E and the scattering angle
q.
Assuming that the surface plane is (001):
- a) Draw the LEED geometry and Ewald sphere, with a plane wave input
beam not necessarily perpendicular to the surface.
- b) Write down an expression for the scattered amplitude from a crystal
into the (hk) reciprocal lattice rod, where the spacings between layers parallel
to the surface are not necessarily equal to the bulk spacing.
- c) Work out the scattered intensity distribution I(V) along the (hk) rod for
the case of normal incidence, where the spacings are equal to the bulk spacing,
and draw the intensity profile.
- d) Show that the peak positions and spacings can be used to calculate the
c-plane spacing, and V0 if f is real. Show that the width of the peaks
is inversely related to l, and hence directly to
V0i.
Problem 3.3: The importance of a high SNR in AES
One of the main problems in Auger electron spectroscopy is that the signal rides
on a non-negligible background, and that the signal to noise ratio (SNR) and the
peak/background ratio (P/B or PBR) can be small. This leads to long data collection
times and/or noisy signals, which are especially troublesome for imaging. The
schemes discussed in section 3.5 (book section 3.5.1) are attempts to approximate
the desired ratio signal with a simple algorithm which can be implemented using
digital data collection and processing. Assuming that the energy channels A, B and
C are equally spaced, with A over the peak, B just above the peak and C an equal
distance to higher energy:
- a) Show that (A-2B+C)/(2B-C) is the simplest linear measure of the P/B ratio,
and that this reduces to (A-B)/B if the background spectrum has zero slope.
- b) Assuming that the measured counts are limited by electron shot noise, find
the SNR of the peak height (A-2B+C) and of the peak to background ratio, explaining
your reasons carefully.
- c) Compare the SNR of the logarithmic measure (A-B)/(A+B) with that of the linear
measure, and convince yourself that it is typically higher for comparable values of
the numbers of counts.