2.11.1 Introduction
When energetic ions are bombarded into crystalline materials, they deposit energy into the target through nuclear and electronic interactions (discussed in detail in Chapter 4). The deposited energy by the ions during these interactions may cause the atoms to be displaced from their lattice sites and cause defects (e.g. vacancies and interstitials). The volume of material disrupted in this way may contain amorphous materials and/or damaged crystalline material. There are many models of ion-beam- induced amorphisation proposed in the literature [129]–[131]. However, ion-beam- induced amorphisation can be attributed to one of the two basic mechanisms (or combination of the two) of damage accumulation – namely homogenous and heterogeneous processes which are discussed below in detail.
2.11.2 Homogenous Amorphisation
Homogenous amorphisation from ion irradiation can be ascribed by a parameter known as critical free-energy, 𝐸𝐸c, or defect density [132]. The defects are built up uniformly throughout the crystal in the irradiated region until it reaches sufficient level to cause a transition from the crystalline state to the energetically favourable amorphous state.
In this model, defects are distributed homogenously in the irradiation-induced collision cascades This damage accumulation increases the free energy of the crystal system as the fluence increases and when the deposited energy rises above a critical value then a spontaneous transition from the crystalline to amorphous state occurs [96]. The 𝐸𝐸c required for amorphisation of Si to occur has been reported as 5×1023 eV.cm–3 [133], [134]. The fluence required for amorphisation based on a critical energy model can be calculated from the known 𝐸𝐸c using Equation 2.8 [132]:
𝜑𝜑𝑋𝑋𝑣𝑣
𝑚𝑚= 𝐸𝐸c
Equation 2.8
where φ (ions.cm–2) is the critical fluence for amorphisation, E
c (eV.cm–3) is the
critical energy density deposited by the ions, Xm (cm) is the depth over which Si is
90% damaged and 𝑣𝑣 (eV) is energy deposited per ion.
However, Baranova et al. [135], [136] suggest in 1975 that point defect density is not sufficient to induce a transition to an amorphous phase but the critical radius, Rcr, of
the damage is also important. If the defect density is higher than the critical defect density in one region but the size of the damaged zone is smaller than Rcr then a
transformation to an amorphous state may occur but it then quickly recrystallises via dynamic annealing. The dynamic annealing can be suppressed at low temperatures and amorphisation can be achieved more rapidly as the defects will be less mobile. In summary, homogenous amorphisation based on a defect density model is dependent on the ion beam parameters. However, other experimental conditions such as sample temperature also play a major role for amorphisation. The Ec required for
amorphisation is strongly dependent on the irradiation conditions and could be achieved quickly at low temperature by suppressing the dynamic annealing [132].
2.11.3 Heterogeneous Amorphisation
In heterogeneous amorphisation, each ion entering the crystal may produce a local amorphous zone with its displacement spike which overlap with zones created by other ions to form a continuous amorphous layer in the crystal [7], [27]–[29].
An ion-beam-induced heterogeneous amorphisation model was proposed by Morehead and Crowder [141] by assuming an amorphous cylindrical core of radius, 𝑅𝑅0, around an ion trajectory within each ion cascade as shown in Figure 2.10:
Figure 2.10: Cylindrical model of radius,𝑅𝑅0, for the volume surrounding the path of an ion track. 𝛿𝛿𝑅𝑅 is the sheath for out-diffusion of defects and 𝑅𝑅𝑜𝑜− 𝛿𝛿𝑅𝑅 is the stable amorphous core. Reproduced from [138].
In this model, R0 is defined by the energy deposited through nuclear collisions per
unit path length. A certain proportion of vacancies or other point defects may diffuse over length Ro in time, t, at temperature, T, thereby reducing the size of the
amorphous core by 𝛿𝛿𝑅𝑅 through dynamic annealing processes. The stable radius of the amorphous core then reduces to 𝑅𝑅𝑜𝑜− 𝛿𝛿𝑅𝑅 . The formation of a continuous amorphous layer proceeds by overlapping of these residual damaged regions. Heavier ions may produce amorphous pockets of large volume (greater R0) within
the cascade and may not require overlapping. The formation of an amorphous layer by overlapping of damage zones can be described by Equation 2.9 based on a kinetic theory described by the Johnson–Mehl–Avrami equation for phase transformations (see Equation 2.10):
𝐹𝐹 = 1 − exp (– 𝑉𝑉. 𝐷𝐷) Equation 2.9
𝐹𝐹 = 1 − exp (– 𝐾𝐾temp𝛷𝛷𝑛𝑛) Equation 2.10
where 𝐹𝐹 is the amorphous fraction; 𝑉𝑉, is the amorphous volume per ion; 𝐷𝐷 is the ion dose (ions.cm–3); 𝑛𝑛 is an exponent describing the 3D growth; 𝛷𝛷 is the fluence (ions.cm–2) and 𝐾𝐾temp is the temperature dependent parameter (cm2.ions–1).
2.11.4 Nucleation Limited Model of Amorphisation
The nucleation limited model suggests a two stage nucleation process for amorphisation: i) production of suitable sites for the accumulation of defects and ii) the interaction of point defects with these sites to produce an amorphous phase [126]. The favourable sites for this process are free surfaces [142], grain boundaries [143] and c–a interfaces [144], [145].
The defect accumulation at these nucleation sites during irradiation may produce complex defects which increase the free energy of the system until the crystal collapses into the amorphous phase [96]. The growth or shrinkage of the amorphous volume depends on the ion fluence and temperature of the substrate. Gibson [105] has extended the Morehead and Crowder [141] model of heterogeneous amorphisation by assuming the overlapping of individual cascades as a requirement for amorphisation as described in Equation 2.11:
𝐹𝐹 = 1 − �(𝑉𝑉. 𝐷𝐷)𝑘𝑘ǃ 𝑘𝑘
𝑚𝑚–1
𝑘𝑘=0
exp (– 𝑉𝑉. 𝐷𝐷) Equation 2.11
where 𝑚𝑚 is the number of overlapping cascades required for amorphisation. For single hit amorphisation within a cascade which requires no overlapping, m = 1 and Equation 2.11 reduces to Equation 2.9.