256 7 Semiconductor surfaces and interfaces

processes can be important in determining outcomes, it is simply not practical to follow them through many layer growth cycles. The approach is to take mesoscopic averages (along steps, or over several layers) and formulate (non-linear) diVerential equations to describe roughening and smoothing and other quantities which can be followed from initial starting conditions. An example is the work of Kardar, Parisi & Zhang (1986), whose KPZ equation describes diVerences from the average thickness h of the layer by a diVusion-like equation, with a nonlinear term proportional to the square of the surface slope, and an added constant, i.e.

-h/-t5n=2h10.5l (=h)21h.

(7.6)

The parameter n describes surface tension which promotes smoothing, l describes the fact that the growth front does not remain planar, and the parameter h (x, t) represents the noise involved in deposition which is random in space and time. Thus solving such equations becomes a series of challenging exercises in applied mathematics and statistical mechanics, which have been extensively described elsewhere (e.g. Krug & Spohn 1992, Barabási & Stanley 1995) and will not be developed further here. Many of these models are, as may be imagined, quite remote from the day to day concerns of the growers of actual devices. This does not mean, of course, that they do not have long, or even medium term signi®cance.

Finally, in concluding this section, we should note that there is an enormous technologically driven thrust behind studies of these device growth processes, which we are only dimly seeing in the more scienti®cally oriented papers which have been highlighted in this section. An example is the eVort to develop blue lasers and high power, high temperature electronics generally, based on the growth of wide band gap semiconductors such as nitrides, e.g. GaN with a band gap of 3.42 eV, and closely related ternary and quaternary compound multilayers. A particular success is the InGaN blue laser diode pioneered by Nakamura. These eVorts are reviewed in several places (e.g. Ponce & Bour 1997, Ambacher 1998, Gil 1998, Nakamura 1998, Hauenstein 1999). It is clear from these accounts that there is much excitement in the ®eld, and that control of growth and doping are central issues which give a lot of diYculty. Some of the background material needed to understand device issues involving thin ®lm and surface processes is given in the next chapter.

Further reading for chapter 7

Ashcroft, N.W. & N.D. Mermin (1976) Solid State Physics (Saunders) chapters 10 and 28.

Davies, J.H. (1998) The Physics of Low-dimensional Semiconductors: an Introduction

(Cambridge University Press), chapters 1±3.

Desjonquères, M.C. & D. Spanjaard (1996) Concepts in Surface Physics (Springer), chapter 5.

Gil, B. (Ed.) (1998) Group III Nitride Semiconductor Compounds (Oxford University Press).

Problems and projects for chapter 7	257

Kelly, M.J. (1995) Low-dimensional Semiconductors (Oxford University Press) chapters 1±3.

Liu, W.K. & M.B. Santos (Eds.) (1999) Thin Films: Heteroepitaxial Systems (World Scienti®c).

Lüth, H. (1993/5) Surfaces and Interfaces of Solid Surfaces (2nd/3rd Edn, Springer) chapter 6.

Mönch, W. (1993) Semiconductor Surfaces and Interfaces (Springer) chapters 7±18. Pettifor, D.G. (1995) Bonding and Structure of Molecules and Solids (Oxford University

Press) chapters 3 and 7.

Sutton, A.P. (1994) Electronic Structure of Materials (Oxford University Press) chapters 6 and 11.

Sutton, A.P. & R.W. BalluY (1995) Interfaces in Crystalline Materials (Oxford University Press) chapter 3.

Tsao, J.Y. (1993) Materials Fundamentals of Molecular Beam Epitaxy (Academic). Yu, P.Y. & M. Cardona (1996) Fundamentals of Semiconductors: Physics and Materials

Properties (Springer) chapter 2.

Zangwill, A. (1988) Physics at Surfaces (Cambridge University Press) chapter 4.

Problems and projects for chapter 7

Problem 7.1. Band structures of Si, Ge and GaAs

Look up the calculated band structures of bulk Si, Ge and GaAs, and explain the meaning of the following terms in relation to these three solids:

(a)the existence of a direct versus an indirect band gap, and the position of the conduction band minimum;

(b)spin-orbit splitting and the diVerences between light and heavy holes;

(c)the removal of degeneracy by stress in compressed thin ®lms, e.g. of Ge on Si(001).

Project 7.2. Band structures in tight binding models

Tight-binding pseudopotential models of tetrahedral semiconductors such as Si or Ge treat the valence s- and p-electrons as moving in the potential ®eld of the nuclei plus closed shell electrons. Consult selected references for section 7.1, and show one of more of the following.

(a)That eight electrons are required to describe the system, resulting in the need to diagonalize an 838 matrix to solve for the band structure as a function of the wave vector k.

(b)That a possible approach to this problem is to use sp3 hybrid states as the basis set, formed by linear combinations of 1 s- and 3 p-electrons as in equation (7.1). Find

the relations between the overlap (or hopping) integrals expressed in the s, px, py, pz system and the sp3 system. What is the potential advantage of the sp3 basis?

(c)That you can construct and solve for the energy bands of Si and or Ge using one

2587 Semiconductor surfaces and interfaces

Table 7.4. Melting temperatures and enthalpies of metals and semiconductors

	Si	Ge	Fe		Ag
Atomic mass (u)	28.1	72.6	55.8		107.9
Tm (K)	1685 6 3.0	1210.4	1809		1234
DHm (kcal/mole)	12.0 6 1.0	8.83	3.3	6 0.1	2.7	6 0.1

or other of these basis sets, and particular values of the matrix elements, using a matrix diagonalization package, display your results graphically, and compare your results to the literature.

(d)That the simplest model of band structure associated with the 231 reconstruction on Si or Ge(001) involves ®xing the atoms below the surface plane in their bulk

positions, and constructing a matrix as a function of wavevectors, kx, ky in the plane of the surface, and kz perpendicular to the surface, again involving an 838 matrix, but now with some of the matrix elements set equal to zero. What is needed

in addition to calculate the equilibrium position of the surface atoms and the resulting surface band structure and surface states?

Problem 7.3. Removal of long range ®elds in polar crystals via surface reconstruction

Draw the planar structure of GaAs to scale, viewed perpendicular to [111], and

(a) identify the planes containing Ga and those containing As, and the [111] versus the

Å Å Å

[111] directions.

If both Ga and As assume the sp3 con®guration,

(b)use Gauss's law to work out the average internal electric ®eld along [111]. Note that a suitable choice of unit cell for the Gaussian surface may convince you that all the long range ®eld is due to the surface layers.

(c)Show that removal of one quarter of the Ga atoms on the (111) A-face (as in ®gure

Å Å Å

7.4) plus the same numbers of As atoms from the (111) B face removes this internal ®eld.

Project 7.4. Adatoms, surface vibrations, roughening and melting transitions on Si and Ge

The melting temperature and latent heat of melting of Si and Ge, Fe and Ag are listed by Honig & Kramer (1969) as in table 7.4.

(a)From these data evaluate the entropy of melting for the four elements, using Appendix C to express the results in units of k/atom.

(b)Do a literature search to ®nd out whether any of the low index surfaces of these

Problems and projects for chapter 7	259

elements are known actually to undergo roughening or melting transitions, and if so under what conditions.

(c)The surface energy of Si(111) has been measured as ,10% less than Si(001) at 600 K, but experiments at T51300 K seem to suggest that their free energies are roughly equal (Bermond et al. 1995). Using the tables in the text, what can you deduce about the diVerence in surface excess entropies between the two faces?

Estimate the surface excess entropy involved (in k/atom units), and compare with the melting entropy, in the following situations, using equations from section 4.2 as appropriate.

(d)We have a certain concentration of ad-dimers on Si(001). Evaluate the con®gurational entropy at T51300 K, assuming that the concentration shown in ®gure 7.10 corresponds to ad-dimers. What is the additional entropy if these ad-dimers are moving around as a 2D gas?

(e)We have asymmetric dimers on Si or Ge(001), each of which can be either up or down with equal probability. Show that this answer must also represent an upper limit to the entropy due to a single step moving freely across the surface. This is a (very rough) approximation to the entropy involved in the roughening transition, discussed in more detail by Desjonquères & Spanjaard (1996, chapter 2).

(f)The amplitude of z-motion of the surface atoms is increased from say 0.01 to 0.05 nm when the p232 or c432 ordered dimer phases on Si(001) disorder into to the `231' (Shkrebtii et al.1995). Is this entropy the same as in (e) or are they additive?