Electron counting rules

Although a number of quantitative surface structure techniques, including sur- face X-ray diffraction [42], ion scattering [43, 44] and low energy electron diffraction [45] have been used in the study of III-V semiconductors surfaces, few provide a direct way of identifying the atomic positions on the surface. A trial structure is often required and the reliable identification of physically valid models from either STM images or LEED patterns is non-trivial. One method for determining the plausibility of a model is through the use of the electron counting rules (ECR) [46]. The dangling bonds that are formed at the surface when a crystal is truncated are energetically unfavourable and so a reduction in the number of dangling bonds will minimise the energy of the surface. The III-V semiconductors achieve this via the transfer of charge to and from dangling bonds, where Group V dangling bonds are filled and Group III dangling bonds are emptied. This charge transfer arises due to the sp3 hybridised bonding orbitals that are present in the zincblende structure and, on this basis, an estimation of the energies of the conduction and valence bands can be made from the energies of the s and p atomic orbitals [46]. In the case of GaAs the Ga dangling bonds lie above the conduction band minimum, and so are emptied, while the As dangling bonds lie below the valence band maximum, and so are filled.

The electron counting rules then ensure that the surface is charge neutral and remains semiconducting as excess charge carries a significant energy cost [47]. Each bond between a Group III and Group V atom consists of two electrons with 2 e− total charge and so, on average, each Group III atom contributes 3₄ e− and each Group V contributes 5₄ e−. Dimers contain 2 e− while charge transfer occurs between the Group III dangling bonds and Group V dangling bonds. To determine if a structure is ECR

[110] [110] Top layer As Second layer Ga Third layer As Fourth layer Ga

Figure 1.5: GaAs(001)-β2(2×4) surface reconstruction, the red dashed line indicates the reconstruction unit cell. Reproduced from Reference [48].

Table 1.2: Example electron counting for the GaAs(001)-β2(2×4) reconstruction as shown in Figure 1.5.

Group III Group V Dimers Group III Group V Total e−

bonds bonds DB DB Charge excess/deficit 3₄ e− 5₄ e− 2 e− 0 e− 2 e− β2(2×4) 12 12 3 4 6 42 Total e− from Ga 12 Total e− from As 30 Total valence 42

compliant a comparison of the total number of electrons available in the reconstruction, found by summing the total number of valence electrons present in non-bulk bonding configurations (3 e− from Group III atoms and 5 e− from Group V atoms), and the number required by the reconstruction is made. The following conditions can be applied:

• Group III bonds contribute 3₄ e−

• Group V bonds contribute 5₄ e−

• Group III dangling bonds have 0 e−

• Group V dangling bonds have 2 e− (from emptied Group III dangling bonds)

• Dimers (and heterodimers) have 2 e−.

Using theβ2(2×4) reconstruction of the GaAs(001) surface, shown in Figure 1.5, as an example then there are 6 As atoms and 4 Ga atoms in non-bulk bonding con-

figurations. Each of these atoms has a single dangling bond and three III-V bonds, while there are three dimers formed between the As surface atoms. A summary of the values determined for this structure are given in Table 1.2 and it can be seen that the number of electrons required matches the number of available valence electrons and so the structure is ECR compliant. Substitution of the atoms within dimers is permitted so long as the atom has only one dangling bond. A Group V atom can be replaced by a Group III atom, which decreases the available number of valence electrons by two but the loss of the filled dangling bond on the Group V then provides two electrons and so accounts for the discrepancy in charge. The reverse is also true and so a Group III atom can replace a Group V atom, both of these processes lead to the formation of heterodimers (mixed Group III-Group V dimers) on the III-V surfaces. It should be noted that while the ECR can be used to predict which surface structures are charge neutral, and so are energetically favourable, they cannot be used to determine the lowest energy structure for a given surface. Typically, experimental results in conjunction with DFT studies are then used to determine the true surface structure. In addition, some experimentally observed structures are known to violate the ECR, in particular on the GaSb, AlSb and GaN surfaces [49].

An extension to the ECR was proposed by Zhang et al. [50] to account for the adsorption of metal adatoms due to their variable valence electron number. These modified rules are known as the generalised electron counting rules (GECR) and they are based on three additional constraints:

1. Metal adatom location:

• d metal adatoms (transition metals) will prefer to occupy interstitial sites

• sp metals will prefer to occupy substitutional sites

2. An adatom will act as a donor (acceptor) if its electronegativity is lower (higher) than those of the component elements of the semiconductor

3. A metal adatom will try to maximise (minimise) its valence electron number if it behaves as an acceptor (donor) in the system.

Accordingly, metal adatoms are able to donate or accept electrons as required and subsequently the surface then reconstructs according to the ECR outlined above. One possible limitation of the GECR arises from the large number of valence electrons of the transition metals. Although, in the case of Mn, it is assumed that adatoms will act as donors (due to its electronegativity relative to that of Ga and As) whereas the situation is more complex for metals which are amphoteric in the semiconductor. An example of this is Fe, which has an electronegativity value between that of Ga and As. It is plausible that Fe could donate its two 4s electrons to the local structure or accept up to four electrons in its 3d subshell and so this mixed behaviour may complicate the determination of the true structural model.