Wood Notation

The reconstruction of a surface results in a new atomic arrangement and a change in the periodicity, defined as the distance between atoms in similar bonding environments, of the surface. One method of relating the periodicity of the bulk structure and the surface periodicity is through the use of matrix notation [40]. If the substrate lattice vectors are defined as aand b and the surface lattice vectors as a0 andb0 then the relationship between the two can be described as follows:

a0 =G11a+G12b

b0 =G21a+G22b

(1.1)

where G is a matrix of the form   G11 G12 G21 G22  . (1.2)

The key advantage of matrix notation is that it allows for the description of incom- mensurate adsorbate layers. A surface structure is incommensurate when there is no simple relation between the adsorbate and the substrate, and they do not share the same translational symmetry.

A more convenient method of representing these changes in surface structure is Wood notation. Here a ratio and a rotation angle are specified between the bulk termination, and the reconstructed surface and is given by

where X is the substrate element or alloy, {hkl} is the surface plane, Rθ is the angle between the substrate and surface lattice vectors, and A is the adsorbate element, if present. The integersnandmare given by|a0| = n|a|and|b0| = m|b|and represent the reconstruction lattice vectors relative to those of the substrate. Note that if there is no rotation in the reconstruction, the -Rθ is omitted, while if the reconstruction is intrinsic to the material and not induced by an adsorbate, then the -A is also omitted. Several prefix letters are often applied to the (n×m) component of the notation in order to remove ambiguity regarding the reconstruction. For primitive or centred reconstruc- tions ‘p’ or ‘c’ are added to denote reconstructions of this form. In this thesis primitive reconstructions have the ‘p’ prefix omitted and so a p(2×2) reconstruction is given as (2×2). Additionally, if the surface exhibits rotational symmetry then it is possible for a reconstruction to be repeated according to this symmetry. In the case of square surface lattices, this rotation is 90◦ and is known as ‘double domain’. For a hexagonal lattice a 120◦ rotation occurs and these are known as ‘triple domain’, and the notation ‘td’ is used.

III-V semiconductor surfaces

The III-V semiconductors are compounds consisting of a Group III element (Al, Ga, In) bonded to a Group V element (P, As, Sb). These alloys are typically seen to have the zincblende structure and, similarly to the Heusler alloys, consist of interpenetrating fcc lattices. The structure features two lattices that are offset by 1₄ of a unit cell relative to each other. Figure 1.4(a) illustrates the zincblende unit cell and it can be seen that each atom forms four bonds with its neighbours in a tetrahedral configuration. When truncating the bulk structure, the bonds between atoms are broken and become partially filled. As the bonding is highly directional in zincblende semiconductors, these unfilled bonds are directed away from the bulk and are known as ‘dangling’ bonds. This is an energetically unfavourable situation and results in neighbouring surface atoms bonding to form dimers.

(b) (c)

Group III (first layer atom) Group V (second layer atom) Group III (third layer atom) (a) [001] [100] [010] [010] [110] [110] [211] [110]

Figure 1.4: Schematics showing the zincblende (a) unit cell (b) (001) surface and (c) (111)A surface, the inset shows a ball and stick model of the GaAs(111) tetrahedral bonding configuration. The red lines and shaded regions indicate the position of the primitive surface mesh and the position of the plane within the unit cell, respectively.

Of relevance to this thesis are the (001) and (111) surfaces which are illustrated in Figures 1.4(b) and (c), respectively. The (001) surface has square symmetry and truncating the bulk in this plane results in the formation of two dangling bonds per atom which, due to the tetrahedral bonding configuration, are aligned along different directions for each atomic species. The Group V bonds are directed along the [110] direction (the n periodicity direction) and the Group III bonds are directed along the [110] direction (the m periodicity direction). In the cubic system the (001), (010) and (100) faces are equivalent by symmetry and in this thesis the convention is made that the growth surface of a cubic substrate is labelled as the (001) face.

The (111) surface is more complex due to the inequivalent number of dangling bonds present depending upon whether it is cation (Group III) or anion (Group V) terminated [41]. Considering a Group III terminated crystal first, an example of which is shown in Figure 1.4(c), the top layer of Ga atoms has one upward dangling bond and three backbonds to the As layer below. Removing the topmost Ga layer exposes the As layer, in which each atom has three upward dangling bonds and a single backbond, which is energetically unfavourable. The As layer then desorbs and a single dangling bond Ga layer remains, this surface is known as the (111)A surface. If, instead, the

crystal is terminated by a layer of As with one upward bond then removal of this As layer exposes a Ga layer with three dangling bonds, which subsequently desorbs resulting in a As terminated layer, this is known as the (111)B surface.