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for example, Sze [101]. The interest here is in the properties of impurity states within semiconductor heterostructures. These states have a bearing on the electronic and optical properties of these materials, and again it is the evaluation of the fundamentals governing their behaviour that is the primary concern. Figure 5.1 shows a schematic representation of a neutral (occupied) donor in a bulk semiconductor. The regular lattice represents a {100} plane. An example of a donor in silicon is the group V element phosphorus. In a compound semiconductor such as gallium arsenide or cadmium telluride, the dopant would occupy a cation site. Thus the atoms of Fig. 5.1 would be gallium or cadmium respectively. In GaAs, a typical donor is silicon, where the latter is forced to sit on the Ga site by growth under an As overpressure. The group IV element Si has one electron more than the group III element Ga and it donates this to the lattice. Note that a Si atom occupying an As site would be an acceptor. It can be seen from the figure that the unionised donor resembles the electronproton pair constituting a hydrogen atom; the physics of this is well understood and was initially described with a semi-classical model by Bohr [4] and reinforced later with full quantum mechanical descriptions [3,29]. The binding energy of this neutral donor electron pair is given by the following:

Like the translation rule, this rule is also a combination of the nesting (11.6) and rearranging rules (11.7) given earlier (because

and the Bohr radius follows as:

o k,p = fok(netok,p)

It is worth listing separately because it is used very frequently. The final rule allows us to 'factor out' multiplication by a constant from a summation. (Conversely, it allows one to 'distribute' multiplication by a constant into a summation.) [Distributivity] (I,k:R:cxT) = cx(Zk:R:T) . (11.16)

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In this case the mass of the donor atom is much larger than the electron mass and hence the simpler 'infinite nucleus mass' variant can be employed. Later, in the chapter dealing with excitons ( 6), this hydrogenic model will again be employed; however, on that occasion the hole mass is of the same order of magnitude as the electron mass and therefore cannot be ignored. Note that the only material parameters needed are the electron effective mass and the permittivity of the host material, and thus the neutral donor binding energy should be independent of the impurity atom. Experimental data reproduced by Sze [44] substantiate this result, for the majority of cases in GaAs. Taking typical values for GaAs, e.g. the F valley electron effective mass me*=0.067 mo and the static dielectric constant er=13.18 [14], then: which is very close to the measured value of 5.8 meV [44], Correspondingly for CdTe, rn*=0.096mo and er=10.6, and therefore: The Bohr radii produced by these simple calculations represent an important result. The lattice constant of GaAs is 5.65 A, and hence 103 A represents around 18 unit cells along any radius from the donor to the electron orbit. Or, alternatively, each face-centred-cubic unit cell contains four lattice points, or eight atoms, and has a volume of A30=(5.65 A)3, and hence the volume occupied by one atom is A30/8. Thus the number of atoms within the spherical electron orbit of radius A is given by:

The general side condition on the application of rules prohibits the use of distributivity when '&' occurs free in the expression c. (Otherwise, any such occurrences would be released/captured by application of the rule.)

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