2246 Electronic structure and emission processes

means of homing in on particular thicknesses which have desired properties. These observations are not only very pretty science, but they hold out the prospect of device applications, such as high density non-volatile memories, and sensitive read/write devices. In particular, the giant magneto-resistance (GMR) and related multilayer eVects are being actively researched, as described in section 8.3.

A ®eld like surface and thin ®lm magnetism, which builds on a long history of electric and magnetic properties, surface physics and growth processes, can be especially diYcult for anyone trying to get started. In this situation, a reasonable initial strategy is to skip all the preliminary work, and go straight to the latest (international) conference proceedings. One conference (from which some examples are taken) was the

Second International Symposium on Metallic Multilayers (MML'95) (Booth 1996); it is especially useful to read the invited papers, since these have more perspective, and typically survey several years of work.

Further reading for chapter 6

Ashcroft, N.W. & N.D. Mermin (1976) Solid State Physics (Saunders College) chapters 8±11, 14, 15 and 33.

Craik, D. (1995) Magnetism: Principles and Applications (John Wiley). Desjonquères, M.C. & D. Spanjaard (1996) Concepts in Surface Physics (Springer)

chapter 5.

Jiles, D. (1991) Magnetism and Magnetic Materials (Chapman and Hall).

Kittel, C. (1976) Introduction to Solid State Physics (6th Edn, John Wiley) chapters 14 and 15.

Pettifor, D.G. (1995) Bonding and Structure of Molecules and Solids (Oxford University Press) chapters 2, 5, 7.

Stanley, H.E. (1971) Introduction to Phase Transitions and Critical Phenomena (Oxford University Press).

Sutton, A.P. (1994) Electronic Structure of Materials (Oxford University Press) chapters 7, 8 and 9.

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

WoodruV, D.P. & T.A. Delchar (1986, 1994) Modern Techniques of Surface Science (Cambridge University Press) chapters 6 and 7.

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

Problems and projects for chapter 6

Problem 6.1. Why is (2kF)21 the characteristic length for electrons at surfaces?

A whole series of problems can be devised to get a feel for the size and relevance of this characteristic length; the following questions require access to a standard solid state textbook (e.g. Ashcroft & Mermin 1976, or Kittel 1976).

Problems and projects for chapter 6	225

(a)For b.c.c Li, ®nd the value of the lattice parameter a, and hence of the nearest neighbor distance. Assuming one electron per atom, evaluate the radius of the

Wigner-Seitz sphere rs, the magnitude of (2kF)21, and hence the periodicity Dz of Friedel oscillations. Indicate these lengths (a and Dz) in relation to ®gure 6.6, and

identify the corresponding periods in the electron density oscillations.

(b)Consider the discussion based on ®gure 6.5, and use this to derive equation (6.4), which is correct asymptotically for z well inside jellium. Note that we cannot use (6.4) near z50. Use the positions of maxima and minima in Lang & Kohn's cal-

culation for rs55 to estimate the value of the phase factor gF, and plot this prediction of n(z) for comparison with ®gure 6.2(a).

(c)Study in outline the ingredients of the Bardeen±Cooper±SchrieVer (BCS) theory of superconductivity, noting how electrons of opposite momenta with energies

within "vD of EF form the BCS ground state, via coupling by phonons of wavevector 2kF.

(d)Similarly note how scattering of electrons across the Fermi surface by phonons of

wavevector 2kF is thought to contribute to structural transitions (charge density or static distortion waves) at metal surfaces, for example in the case of Mo(001).

(e)Estimate the wavelength of the ripples seen in the quantum corral of ®gure 6.4, and relate this length to the Fermi surface of copper, incorporating the cylindrical geometry. Consult the paper by Petersson et al. (1998) to see how such measures can be made quantitative.

Problem 6.2. Derivation of the Richardson±Dushman equation from free electron theory

Consider the Fermi±Dirac distribution in the form f(E)5(11exp(E2 m)/kBT)21, where the chemical potential of the electron gas, m is the same as the Fermi energy EF.

(a)Use this form, together with the density of states shown in ®gure 6.5 and the

concept of the perpendicular energy, Ez5("kz )2/2m, to derive the free electron form of the Richardson±Dushman equation (6.7) when barrier transmission occurs for all Ez.m.

(b)By performing a 1D calculation, matching electron waves at the surface at z50, investigate how the re¯ection coeYcient for electrons incident on the surface

changes as a function of Ez when Ez $ m. Use this result to show how energy-depen- dent barrier transmission aVects the formula derived in part (a).

Project 6.3. Barrier transmission, the Fowler±Nordheim equation and models of STM operation

The Fowler±Nordheim equation can be derived from free electron theory in a similar manner to problem 6.2, except that we now need the probability of transmission through the ®nite barrier as a function of Ez. This is typically given by the Wentzel±Kramers±Brillouin (WKB) approximation, where the transmission coeYcient

2266 Electronic structure and emission processes

T,exp[2 (2/")e(V(z)2 Ez)1/2dz]. In this formula, the potential V(z) is as in ®gure 6.11, and the limits of the z-integration are set by the perpendicular energy Ez.

(a)Show that the result (6.8) arises in the limit of a triangular barrier at zero temperature, and that the energy distribution will be broadened at elevated temperature as in ®gure 6.12.

(b)Show that the same set of arguments applied to the operation of the STM, predict qualitatively the observed exponential dependence of tip-sample voltage on tip separation at constant tunneling current, in the limit of small voltages.

(c)Alternatively, you may prefer to start from the quantum-mechanical expression for the particle current in terms of gradients of the (1D) wavefunction, and show using ®rst order perturbation theory that the tunneling current I is given by

I5(e/") omn [ f(Em)2 f(En)] |Mmn |2d(En 1V2 Em),

where the matrix element Mmn is given by

Mmn 5("/2m)edS´(cm*=cn 2 cn=cm*),

where dS is the element of surface area lying in the barrier region. In these formulae, f(E) is the Fermi function and d(E) the Dirac d-function.

Project 6.4 Interactions between d-electrons, magnetism and structures in transition metals

Starting from the relevant sections of Sutton (1994) and/or Pettifor (1995), and other literature cited in section 6.3.4, investigate models of b.c.c. and f.c.c. Fe, and neighboring elements in which you are interested.

(a)Describe how magnetism stabilizes the b.c.c. structure at low temperature, and ®nd out about f.c.c. Fe at high temperatures, and the magnetism of neighboring elements in the 3d series.

(b)What features of the 4d and 5d series make the elements in corresponding columns of the periodic table non-magnetic?

(c)Investigate ideas of bond-order potentials, and describe how `embedding bonds' can create angular dependent interactions, and thus help to stabilize crystal structures which are not close-packed.

(d)Show that the angular dependent interactions can be formulated in terms of higher

moments (mi with i.2) of the valence electron density of states, and that the b.c.c. stability is largely attributable to m4, while the h.c.p.±f.c.c. diVerence is aVected by m6. From this viewpoint, investigate the contribution such eVects can make to the surface energy of transition metals.

7Semiconductor surfaces and interfaces

This chapter gives a description of semiconductor surfaces, and the models used to explain them. Section 7.1 outlines ideas of bonding in elemental semiconductors, and these are used to discuss case studies of speci®c semiconductor surface reconstructions in section 7.2, building on the survey given in section 1.4. If you are not familiar with semiconductors and their structures, you will also need access to sources that describe the diamond, wurtzite and graphite structures, and which also describe the bulk band structures; these points can be explored via problem 7.1. It is also very helpful to have some prior knowledge of the terms used in covalent bonding, such as s and p bands, sp2 and sp3 hybridization. Section 7.3 describes stresses and strains at surfaces and in thin ®lms, including the thermodynamic discussion delayed from section 1.1; the importance of such ideas in the growth of semiconductor device materials is discussed, especially those based on the elements germanium and silicon, with references also given to the 3±5 compound literature.

7. 1. Structural and electronic effects at semiconductor surfaces

The ®rst thing to realize is that the reconstructions of semiconductor surfaces are not, in general, simple. In section 1.4 reconstructions were introduced via the (relatively simple) Si(100) 231 surface. This introduced ideas of symmetry lowering at the surface, domains, and the association of domains with surface steps. At the atomic cell level we saw the formation of dimers, organized into dimer rows. If all this can happen on the simplest semiconductor surface, what can we expect on more complex surfaces? More importantly, how can we begin to make sense of it all? This is a topic which is still very much at the research stage. But enough has been done to try to describe how workers are going about the search for understanding, which is what is attempted here.

7.1.1Bonding in diamond, graphite, Si, Ge, GaAs, etc.

The basis of understanding surfaces comes from considering them as intermediate between small molecules and the bulk. In the case of the group 4 elements, there is a progression from C (diamond, with four nearest neighbors), through Si and Ge with the same crystal structure, then on to Sn and Pb. The last two elements are metallic at

227

228 7 Semiconductor surfaces and interfaces

Energy,

Ep

Es

(a)

sp3 antibonding

conduction band

p band

s band

sp3 bonding valence band

Sn Ge Si

C

Bond integral (h)

Figure 7.1. (a) Hybridization gap in due to sp3 bonding in diamond, Si, Ge and gray Sn; (b) stages in the establishment of the valence and conduction bands via s-p mixing, involving DEsp and the overlap integral h (after Harrison 1980, and Pettifor 1995, replotted with permission).

room temperature, Pb having the `normal' f.c.c. structure with 12 nearest neighbors. We might well ask what is giving rise to this progression, and where Si, Ge, GaAs, etc. ®t on the relevant scale. A frequent answer is to say something about sp3 hybrids, assume that is all there is to say, and move on. However, there is much more to it than that; the extent to which one can go back to ®rst principles is limited only by everyone's time (Harrison 1980, Sutton 1994, Pettifor 1995, Sutton & BalluY 1995, Yu & Cardona 1996).

In lecturing on this topic, I have typically started with a two-page handout, the essence of which is given here as Appendix K. This connects bonding and anti-bonding orbitals in s-bonded homonuclear diatomic molecules with the overlap, or bonding integral, h. (Note that h is not Planck's constant, and the symbol often used for overlap integral is b which is not (kT)21.) For heteronuclear diatomic molecules where DE is the energy diVerence of levels between the molecules A and B, the splitting of the levels wAB combines as

w	AB	5Î(4h2	1DE2).	(7.1)

This leads to ideas, and scales, of electronegativity/ionicity, based on the relevant value of (DE/h): for group IV molecules this is zero, increasing towards III±V's, II±VI's etc., roman numerals being the convention for the diVerent columns of the periodic table; these scales try to establish the relevant mixture of covalent and ionic bonding in the particular cases: 3±5's are partly ionic, and 2±6's are clearly more so.

In the diamond structure solids, the tetrahedral bonding does indeed come from sp3 hybridization, but it is not obvious that this will produce a semiconductor, and the question of the size of the band gap, and whether this is direct or indirect, is much more subtle, as indicated in ®gure 7.1. The s-p level separation in the free atoms is about 7±8 eV, but the bonding integrals are large enough to enforce the s-p mixing and to open