Problems for chapter 1	33

1.5.7Screening

The above description emphasizes the importance of screening, in general, and also in connection with surfaces. We can also notice the very diVerent length scales involved

in screening, from atomic dimensions in metals, (2kF)21, 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. in a plasma). In general, nature tries very hard to remove long range (electric and magnetic) ®elds, 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.

Further reading for chapter 1

Adamson, A.W. (1990) Physical Chemistry of Surfaces (Wiley, 5th Edn) chapters 2 and 3.

Blakely, J.W. (1973) Introduction to the Properties of Crystal Surfaces (Pergamon) chapters 1 and 3.

Clarke, L.J. (1985) Surface Crystallography: an Introduction to Low Energy Electron DiVraction (John Wiley) chapters 1, 2 and 7.

Desjonquères, M.C. & D. Spanjaard (1996) Concepts in Surface Physics (Springer) chapter 1.

Gibbs, J.W. (1928, 1948, 1957) Collected Works, vol. 1 (Yale Univerity Press, New Haven); reproduced as (1961) The Scienti®c Papers, vol. 1 (Dover Reprint Series, New York).

Henrich, V.E. & P.A. Cox (1996) The Surface Science of Metal Oxides (Cambridge University Press) chapters 1, 2.

Hudson, J.B. (1992) Surface Science: an Introduction (Butterworth-Heinemann) chapters 1, 3±5, 17.

Kelly, A. & G.W. Groves (1970) Crystallography and Crystal Defects (Longman) chapters 1±3.

Lüth, H. (1993/5) Surfaces and Interfaces of Solid Surfaces (3rd Edn, Springer) chapters 3, 6.1, 6.2.

Prutton, M. (1994) Introduction to Surface Physics (Oxford University Press) chapters 3 and 4.

Sutton, A.P. & R.W. BalluY (1995) Interfaces in Crystalline Materials (Oxford University Press) chapter 5.

Problems for chapter 1

These problems test ideas of bond counting, elementary statistical mechanics, diVusion and surface structure. When set in conjunction with a course, 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. Note that there

341 Introduction to surface processes

are further problems of a similar type in Desjonquères & Spanjaard (1996, chapters 2 and 3).

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

Consider the (012) face on a Kossel (simple cubic) crystal with six 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 ®gures 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. There is a more general vector-based approach by MacKenzie et al. (1962), but this is not simple for a ®rst try, or for complex structures. If the stereograms (®gure 1.8) are not familiar, see Kelly & Groves (1970) or another crystallography book, or obtain web-based information via Appendix D.

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

Consider the (001) face of an 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)Construct a diVerential equation to describe the processes of arrival of atoms from, the and re-evaporation into the vapor, to ®nd the equilibrium concentration of adatoms in monolayer (ML) units. Find the adatom concentration at T51000 K if the arrival rate R51 ML/s.

(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 (a) corresponds to underor 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 ®gure 1.9. This model neglects the eVects of vacancies, and the model of lattice vibrations is only a ®rst approximation.

Problems for chapter 1	35

(a)How might you consider the eVects 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 eVect of other lattice dynamical models, for example the cell model, discussed in more detail in chapter 4.2?

Note: this problem is useful for a discussion of points of principle and practicality, and could be expanded via detailed computation for a course project.

Problem 1.4. Crystal growth at steps and the condensation coef®cient

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

(a)Set up a one-dimensional rate-diVusion equation describing the diVusion 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 pro®le of adatoms between the steps depends on the

ratio cosh (x/xs)/ cosh(d/2xs), where xs is the BCF length (section 1.3). Show that the fraction of atoms which get incorporated into the steps, the condensation

coeYcient, is given by (2xs /d)tanh(d/2xs). Evaluate the limits (2xs /d) ..1 and ,, 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 231, and 231-like structures, or in semiconductors the diVerence between 231, c432 and p 232 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 diVraction pattern, most obviously the LEED patterns in the literature.