PHY 598: Surfaces and Thin Films (Venables)
Assignments for Section 1, Spring 2007

The first and second assignments are due to be completed before Spring break, but I will set individual deadlines before then by email. I expect three of the following problems to be done by the agreed deadline. There will be a second assignment on section 2 and parts of 3. Please let me know which subjects interest you as a topic for a talk or mini-project for later in the semester.

These problems test ideas of bond counting, elementary statistical mechanics, diffusion and surface structure. They have typically not been done ‘cold’, but have been used to open a discussion on topics which are best attempted through problem solving rather than by lecturing. There are further problems of a similar type in Desjonquères and Spanjaard, chapters 2 and 3. Make sure that you see me in time if you need some help- you are not expected to be able to do this assignment without some assistance.

• Problem 1.1: Bond counting and surface (internal) energies of a static lattice

  Consider the (012) face on a Kossel (simple cubic) crystal with 6 nearest neighbor bonds.

  a) Use the analysis of section 1.2 to consider the surface energy of this crystal in terms of the sublimation energy L and the lattice parameter a. Find the ratio of the surface energy of the (012) and the (001) face.

  b) Repeat this exercise for the (012) face of a f.c.c crystal with 12 nearest neighbor bonds. Compare your result with diagrams 7 and 9 (book figures 1.6 and 1.8), and comment on the relative values.

  Note: this problem can be done most readily by drawing the structure and counting bonds. However, there is a general approach you could consider, described by MacKenzie et al. (1962). If the stereograms shown in figure 1.6 are not familiar, consult a crystallography book such as Kelly and Groves (1970), or use the web-based book comments for section 1.3 or go directly to a 1998 Sussex student project on the Properties of stereograms.

• Problem 1.2: Local equilibrium at the surface of a crystal at temperature T

  Consider the (001) face of a f.c.c crystal with 12 nearest neighbor bonds, and (small concentrations of) adatoms and vacancies at this surface. The sublimation energy is 3eV and the frequency factor is 10 Thz. Use the appropriate formulations of section 1.3 to do the following:

  a) Express the local equilibrium between adatom evaporation and the rate of arrival, R, of atoms from the vapor, to find the concentration of adatoms in monolayer (ML) units. Find the adatom concentration at 1000K if R = 1 ML/sec.

  b) Use the chemical potential formulation to express the local equilibrium between the bulk crystal and the surface adatoms, to obtain their equilibrium concentrations at the same temperature, ignoring arrival from, or sublimation to the vapor. Hence decide whether the case described in a) corresponds to under- or over-saturation, and calculate the thermodynamic driving force in units of kT.

• Problem 1.3: Effects of vacancies and/or lattice vibrations on the sublimation pressure

  Consider the model of the vapor pressure of a solid described in section 1.3, table 1.1 and diagram 10 (book figure 1.9). This model neglects the effects of vacancies, and the model of lattice vibrations is only a first approximation.

  a) How might you consider the effects of vacancies, which are expected to have an energy of 1 eV in Ag, and reduce the frequency of atomic vibration in the vicinity of the neighbors of the vacancy to 80% of the value in the bulk.

  b) How might you consider the effect of other lattice dynamical models, for example the cell model, discussed in more detail later in section C2 (book section 4.2).

  Note: this problem is useful for a discussion of points of principle and practicality, and can be expanded via detailed computation for a course project. The Effects of Vacancies were considered by Bala Ramadurai with an interactive Javascipt/Java program in 1999. This has now disappeared from the web, so we could do it again!

• Problem 1.4: Crystal growth at steps and the condensation coefficient

  Consider a surface consisting of terraces of width d, separated by monatomic height steps.

  a) Set up a 1-dimensional equation describing the diffusion of adatoms to the steps in the presence of both adatom arrival and desorption. Explain what boundary conditions you use at the steps.

  b) Show that the steady state profile of adatoms between the steps depends on the ratio cosh (x/X)/ cosh(d/2X), where X is the BCF length (section 1.3). Show that the fraction of atoms which get incorporated into the steps, the condensation coefficient is given by (2X/d)tanh(d/2X). Evaluate the limits (2X/d) when much greater than 1 and much less than 1, and give reasons why these limits are sensible.

• Problem 1.5: Surface reconstructions of particular crystals

  Consider a surface structure in which you are interested. In metals this could be W and Mo(100) which have transitions below room temperature to 2x1, and 2x1-like structures, or in semiconductors the difference between 2x1, c4x2 and p2x2 superstructures on Si or GaAs(100).

  Use your chosen system to explore the relation between the structure, the symmetry and size of the surface unit cell, and the diffraction pattern, most obviously the LEED patterns in the literature.

  Note: if you want to study a more complex structure on a lower symmetry substrate, have a discussion first so that you don’t spend your time unprofitably.

• Problem 1.6: Image forces, screening and the work function

  In the book, page 32, section 1.5.6, I have written, perhaps too glibly, that V(z) = -e/4z for the potential due to the image charge, which is reduced by the factor (e-1)/(e+1) for a dielectric with dielectric constant e.

  a) What units are these formulae expressed in, and what would the MKS (SI) formulae be? Explain why and when this formula becomes unrealistic as the distance z tends to zero.

  b) Work out the size of the image potential in Volts at distances of 0.1, 0.5 and 1 nm (or produce a graph) for (e = infinity, 10 and 2.

  In the book, pages 186 (table 6.1) and 194 (table 6.2), the work function values are given in eV. And on pages 200-203 there is a discussion of thermionic and field emission involving the work function.

  c) Is the work function a voltage or an energy? Do you think the discussion on page 193-195 is consistent, or should it be clarified, and if so, how?

  d) If we decided that the work function is an energy, would the units of B in the Fowler-Nordheim equation (6.8) on page 202 need to be changed, and if so how?