Superlattice

Figure A2.5.18. Body-centred cubic arrangement of (3-brass (CiiZn) at low temperature showing two interpenetrating simple cubic superlattices, one all Cu, the other all Zn, and a single lattice of randomly distributed atoms at high temperature. Reproduced from Hildebrand J H and Scott R L 1950 The Solubility of Nonelectrolytes 3rd edn (New York Reinliold) p 342.

The treatment of such order-disorder phenomena was initiated by Gorsky (1928) and generalized by Bragg and Williams (1934) [5], For simplicity we restrict the discussion to the synnnetrical situation where there are equal amounts of each component (x = 1/2). The lattice is divided into two superlattices a and p, like those in the figure, and a degree of order s is defined such that the mole fraction of component B on superlattice p is (1 +. s)/4 while that on superlattice a is (1 -. s)/4. Conservation conditions then yield the mole fraction of A on the two superlattices... 

If the entropy and the enthalpy for the separate mixing in each of the half-mole superlattices are calculated and then combined, the following equation is obtained ... 

Ghahramani E, Moss D J and Sipe J E 1990 Second-harmonic generation in odd-period, strained, (Si)y(Ge) Si superlattices and at Si/Ge interfaces Phys. Rev. Lett. 64 2815-18... 

One fiirther method for obtaining surface sensitivity in diffraction relies on the presence of two-dimensional superlattices on the surface. As we shall see fiirtlrer below, these correspond to periodicities that are different from those present in the bulk material. As a result, additional diffracted beams occur (often called fractional-order beams), which are uniquely created by and therefore sensitive to this kind of surface structure. XRD, in particular, makes frequent use of this property [4]. Transmission electron diffraction (TED) also has used this property, in conjunction with ultrathin samples to minimize bulk contributions [9]. 

The balance between these different types of bonds has a strong bearing on the resulting ordering or disordering of the surface. For adsorbates, the relative strength of adsorbate-substrate and adsorbate-adsorbate interactions is particularly important. Wlien adsorbate-substrate interactions dominate, well ordered overlayer structures are induced that are arranged in a superlattice, i.e. a periodicity which is closely related to that of the substrate lattice one then speaks of commensurate overlayers. This results from the tendency for each adsorbate to seek out the same type of adsorption site on the surface, which means that all adsorbates attempt to bond in the same maimer to substrate atoms. 

A superlattice is temied commensurate when all matrix elements uij j are integers. If at least one matrix element uij j is an irrational number (not a ratio of integers), then the superlattice is temied incommensurate. A superlattice can be inconnnensiirate in one surface dimension, while commensurate in the other surface dimension, or it could be mconmiensurate in both surface dimensions. 

A superlattice can be caused by adsorbates adopting a different periodicity than the substrate surface, or also by a reconstmction of the clean surface. In figure B 1.21.3 several superlattices that are conmionly detected on low-Miller-index surfaces are shown with their Wood notation. 

Figure Bl.21.3. Direct lattices (at left) and corresponding reciprocal lattices (at right) of a series of connnonly occurring two-dimensional superlattices. Black circles correspond to the ideal (1 x 1) surface structure, while grey circles represent adatoms in the direct lattice (arbitrarily placed in hollow positions) and open diamonds represent fractional-order beams m the reciprocal space. Unit cells in direct space and in reciprocal space are outlined.

So it is essential to relate the LEED pattern to the surface structure itself As mentioned earlier, the diffraction pattern does not indicate relative atomic positions within the structural unit cell, but only the size and shape of that unit cell. However, since experiments are mostly perfonned on surfaces of materials with a known crystallographic bulk structure, it is often a good starting point to assume an ideally tenuinated bulk lattice the actual surface structure will often be related to that ideal structure in a simple maimer, e.g. tluough the creation of a superlattice that is directly related to the bulk lattice. 

With some practice, one can easily recognize specific superlattices from their FEED pattern. Otherwise, one can work tluough the above equations to connect particular superlattices to a given FEED pattern. A number... 

Islands occur particularly with adsorbates that aggregate into two-dimensional assemblies on a substrate, leaving bare substrate patches exposed between these islands. Diffraction spots, especially fractional-order spots if the adsorbate fonns a superlattice within these islands, acquire a width that depends inversely on tire average island diameter. If the islands are systematically anisotropic in size, with a long dimension primarily in one surface direction, the diffraction spots are also anisotropic, with a small width in that direction. Knowing the island size and shape gives valuable infonnation regarding the mechanisms of phase transitions, which in turn pemiit one to leam about the adsorbate-adsorbate interactions. 

Bartlett P, Ottewill R FI and Pusey P N 1992 Superlattice formation in binary mixtures of hard-sphere colloids Phys. Rev. Lett. 68 3801-4... 

Andres R P ef a/1996 Self-assembly of a two-dimensional superlattice of molecularly linked metal clusters Science 273 1690... 

Flarfenist S A and Wang Z L 1999 Fligh-temperature stability of passivated silver nanocrystal superlattices J. Phys. Chem. B 103 4342... 

Sun S and Murray C B 1999 Synthesis of monodisperse cobalt nanocrystais and their assembly into magnetic superlattices J. Appl. Phys. 85 4325... 

Fig. 8. Giant negative magnetoiesistance RjR H = 0) behavioi of three Fe/Ci superlattices as a function of applied field at 4.2 K showing the value of the complete switching field, for each lattice. The current and the appHed field are along the (110) direction ia the plane of the layers (60). To convert T to...

Heterostructures and Superlattices. Although useful devices can be made from binary compound semiconductors, such as GaAs, InP, or InSb, the explosive interest in techniques such as MOCVD and MBE came about from their growth of ternary or quaternary alloy heterostmctures and supedattices. Eor the successful growth of alloys and heterostmctures the composition and interfaces must be accurately controlled. The composition of alloys can be predicted from thermodynamics if the flow in the reactor is optimised. Otherwise, composition and growth rate variations are observed... 

Eig. 4. A transmission electron photomicrograph of an InAsSb—InSb superlattice grown by MOCVD. 

Surface phase transformations and surfrice chemical reactions are followed by studying the time evolution of superlattice beams originating from monolayer or submonolayer films. See, for example. Chapters 8-10 in Low-EnergyYaj Hove et al. op cit.). 

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