Critical density model

One of the main mechanism explaining amorphisation is based on the density of the defects induced during ion irradiation within the region made amorphous and has been developed via the so-called critical density model, which as will be addressed later, is defined as a homogeneous amorphisation mechanism [23], [26], [27], [127]. According, to this model, when defects accumulate in the crystal there is a certain density at which the crystal should collapse into its amorphous phase [23], [26], [27], [127]. Through irradiation, the Gibbs free energy of the irradiated material increases, thus making it energetically favourable for the

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material to be amorphous rather than to be a highly damaged crystal [23], [26], [27], [127]. This model has been developed mainly for silicon in order to predict the appearance of an amorphous layer during implantation as well as the depth of that layer [26].

In line with this, Swanson et al. have proposed that the threshold density of defects in silicon and germanium should be around 0.02 of the atomic fraction [138]. In semiconductors, the density of defects is considered proportional to the energy absorbed in the crystal via nuclear collisions [27]. Therefore, rather than the density of defects, in the critical energy density model the nuclear energy deposited in the target material is instead used to determine when and at which depth amorphisation will occur [23], [26], [27], [127]. As these absorbed energies can be calculated via software such as SRIM (as for instance shown in figure 2.16), the critical density model has been used in the literature to predict the formation of amorphous layers in both silicon and germanium [23], [26].

Figure 2.16: Nuclear energy absorbed by the target material as a function of depth in

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In the critical density model, the strong assumption is that the energy density is homogenous within the damaged layer. This assumption is often disputed as the energy distribution during irradiation depends not only on the overall energy deposited in the crystal but also on the morphology of the collision cascade which might induce localised regions with higher concentrations of defects, whilst others regions of the irradiated region might be defect-free [27], [116]. However, some authors have stated that a homogenous amorphisation mechanism may still hold true as the concentration of defects can become independent of the morphology of the collision cascades if Frenkel pairs diffuse and become homogenously distributed in the irradiated regions [27]. In such a model, under conditions where dynamical annealing (i.e. annealing effects occurring during damage build-up via the ion beam) can be neglected, the critical energy density at which amorphisation occurs should therefore remain identical irrespective of the ion.

Using the critical dose energy density (Edc) at which germanium becomes amorphous, the depth

and position of the amorphous layer can be determined as a function of fluence using equation 2.15, where ψ is the fluence, 𝐸𝑑(𝑧) is the energy absorbed by the target material at a depth z and ρ is the atomic density of the target material [26].

𝜓. 𝐸_𝑑(𝑧) = 𝐸𝑑𝑐. 𝜌 (2.15)

In germanium, at RT, self-ion irradiation performed at 150 keV has been correlated to the depth, z, of the amorphous layer using cross section TEM (X-TEM) in [26]. Using equation

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2.15, the correlation between critical fluence and the amorphisation depth allowed Claverie et al. to conclude that the threshold energy density (Edc) was 5 eV per atom in germanium [26].

Furthermore, as shown in figure 2.17 the authors analysed results from other publications on the irradiation of germanium by phosphorus, germanium and indium ions at RT as well as irradiations performed by phosphorus, germanium, arsenic and bismuth ions at liquid-nitrogen temperature. They concluded that in all those experiments, the same Edc was shared as the

predicted amorphous depth corresponded to what could be expected using an Edc of 5 eV. They

therefore concluded that these results confirmed that the density of defects was irrespective of the bombarding ion. Furthermore, as RT and liquid nitrogen experiments were equivalent in term of critical energy density, it was also concluded that at RT under the conditions used in these experiments the dynamical annealing was low enough to have little effect on the overall defect density [26].

Figure 2.17: Measured amorphous depth due to irradiation of germanium as reported in

various experiments reported in the literature and summarised in [26] to be compared with the predicted amorphous depth if Edc = 5eV. From [26].

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However, the authors expressed reservations concerning a value Edc of 5 eV for lighter ions

[26]. As amorphisation by light ions has been shown to be difficult or even impossible, the authors irradiated germanium with boron at RT and at liquid nitrogen temperature. They observed that a reduced amorphisation layer was induced by the ion beam compared with that predicted by the Edc calculated with heavier ions. Whilst the authors did not explicitly quantify

the reduction in the amorphisation layer compared with the model and the value Edc value of 5

eV, they explained that the discrepancy was due to enhanced Frenkel pair recombination in very dilute cascades. It is also worth noting that unlike in the work of Claverie et al. work (in [26]) the effect of temperature on the density of defects during self-ion irradiation has been reported elsewhere as being an important factor as Edc = 5 eV at RT but only 1.2 eV when the

irradiation is performed at liquid nitrogen temperature [23].