Continuum models (Sec. 2.2) were developed to describe the relaxation times of electron-ion non-equilibrium. These models were traditionally used to model ultrashort pulse laser illu- mination and later swift heavy ion impacts. On the other hand, primary radiation damage description (caused, say, by high-energy neutrons) requires atomistic models and specifically the knowledge of the number of defects produced after an irradiation event. The first model- ling attempts to calculate this involved the binary collision approximation (Sec. 2.3.1). This method was later slowly taken over by classical molecular dynamics, which offered a better parameterisation of the atomistic interactions. However, it wasn’t until the beginning of the century when the electron-ion non-equilibrium ideas and molecular dynamics were merged into hybrid (also called augmented) models (Sec. 2.3.2). More sophisticated modelling devel- opments in radiation damage involve incorporation of the electron-ion non-equilibrium caused by irradiation by simulating the electron and ion dynamics explicitly (Sec. 2.3.3).
2.3.1 The binary collision approximation
The binary collision approximation (BCA) model simulates a cascade event, initialised by an energetic pka, simply as a series of collisions involving two atoms only. Atoms are assumed to travel in straight lines at constant velocity between the collisions. When atoms come within a
2.3. Atomistic models
given distance, they interact and their new directions and kinetic energies are determined from scattering theory. In the BCA model, since there are only four types of possible scattering out- comes (after [21]), which depend on the kinetic energy transferred to the target (Et), remaining kinetic energy of the projectile atom (Ep), spherically-averaged threshold displacement energy (TDE) Ed (i.e. the energy required to permanently displace an atom from its lattice site) and
a small energy cutoff Ecut:
1. Et> Ed, Ep > Ecut: The projectile atom continues colliding and the target atom joins
the cascade,
2. Et > Ed, Ep < Ecut: The projectile atom scatters off the target atoms, which remains
on its lattice position,
3. Et< Ed, Ep > Ecut: The projectile atom replaces the target at its lattice site, and the
target atom joins the cascade instead,
4. Et < Ed, Ep < Ecut: The projectile atom becomes a local interstitial, while the target
atoms remains on its lattice position.
The SRIM [54] package is an example program, which uses the above algorithm to calculate the range of a pka in a given target. Additionally, TRIM [61], which is at the core of SRIM and precedes it, gives the resultant distributions of ions and damage yield. Another popular code, MARLOWE [62] is commonly used in BCA simulations in crystalline, rather than amorphous, targets. Later version of the codes additionally included the effects of electronic stopping in the formalism of Firsov [22], and Lindhard and Scharff [23].
The BCA model has been used to study collision cascades [63, 64] and sputtering yields [65, 66]. This method was the basis for the famous “NRT” (after Norgett, Robinson and Torrens [67] and Kinchin and Pease [68]) empirical formula, which related the number of Frenkel pairs NF P to the damage energy in a linear fashion (equivalent here to the pka energy Epka as the
electronic effects are neglected):
NF P =
0.8Epka
2Ed
. (2.17)
The method, despite setting a standard for a long time, omits a lot of relevant physics. First of all, it does not include the many-body effects, which at higher pka energies would most certainly happen. Because of that it does not predict a local transient melt phase or a thermal spike in cascade collisions. Secondly, because of a lack of any atomistic dynamics apart from the simple binary collision algorithm, it cannot describe defect diffusion and recombination occurring at the primary radiation damage formation stage.
2.3.2 Molecular dynamics
An explicit representation of ions and inclusion of many-body effects in MD marks a significant improvement over the BCA model. Due to the growth in computational power and development of parallel MD codes, studying primary radiation damage with MD has gradually become more
popular. Earliest MD studies (with low pka energies of 400 eV) date back to 1960s (Gibson, Goland, Milgram and Vineyard [69]) and were among the first applications of MD - the first use of the MD technique was reported in 1959 [70]. However, it wasn’t until the 1990s when large-scale simulations with complex many-body potentials became affordable that MD became a standard tool for primary radiation damage study.
The current achievement of classical cascade MD simulations, as applied to radiation dam- age problems has been extensively reviewed, for instance, in [71–73]. In brief, cascade MD simulations have shown that (after [71]): (i) cascade recombination effects absent in the BCA reduce the amount of the residual damage to a fraction (0.3) of the NRT predictions, (ii) high- energy cascades break up into sub-cascades and this branching process boosts recombination, (iii) large clusters of SIA can already be formed by primary radiation damage processes.
Below we concentrate on the MD-based modelling methods, which relax the assumption of local electron-ion non-equilibrium and attempt to include a degree of electronic excitation effects.
2.3.2.1 Augmented molecular dynamics
The augmented MD methods aim to include the effects of electronic stopping and electron- phonon coupling as additional damping and driving terms in the atomistic equations of motion. The electronic stopping effect for slow and heavy particles can be included in molecular dy- namics as an additional friction force proportional to the projectile velocity using the fact that Se∝ v. Caro and Victoria [74] and Finnis, Agnew and Foreman [75] have pioneered the idea
of viscous MD with a non-directional damping term applied to all ions above an energy cutoff Ec- an ad hoc value that prevents an equilibrium system (or later stages of a cascade simula-
tion) from being artificially damped. This approach has become almost a standard method of extracting the energy due to electronic stopping in molecular dynamics cascades and has been used extensively since.
Such an approach assumes that the local electronic temperature rise does not have an impact on the dynamics and that this energy is not fed back to the atoms. The original approach by Caro and Victoria [74] also included an additional sink term for the atoms that represented the electron-phonon coupling. The sink was linked to a target temperature and neglected any local heating effects. The local effects of electron-phonon coupling were pioneered by Duffy and Rutherford [76] via the so-called inhomogeneous Langevin thermostat mechanism. Such inhomogeneity necessitated the inclusion and concurrent tracking of a local electronic temperature in a two-temperature formalism.
The effect of electron-phonon coupling on primary radiation damage was touched upon in several papers ([77–79] and more recently in [80]). It was concluded that the strength of the e-p coupling has a significant effect on the primary damage formation - it can increase [78] or, under some circumstances, decrease the number of defects [81]. Nonetheless, it is still uncertain how to include the effect of electron-phonon coupling in molecular dynamics in a precise physical manner. The method we adapt here, which broadly follows [76], is presented in Sec. 4.3.
2.3. Atomistic models
2.3.2.2 Including excited state interatomic potentials
Molecular dynamics with its augmented applications to non-equilibrium dynamics is built on the assumption that the interatomic potentials remain unmodified by the strong local electronic excitations. This can be a good approximation to some extent and we will revisit this point in chapter 6. However, in general excitations of the order of several eV can cause bond weakening, bond breaking and other more complex effects on the potential energy surface (some of which are discussed in Sec. 3.3.2.3).
Some early attempts at including the change in the bonding character involved transient and local removal of the attractive part of the potential [82, 83]. This has obvious limitations, as the resulting excited-state potential is entirely phenomenological and not constructed to reproduce any particular property. An empirical fitting procedure for potentials dependent on an effective electronic temperature was presented in [84] for tungsten and in [85] for silicon. An alternative scheme based on the force-matching technique [86] was applied to construct a Te-dependent potential for gold [87]. To date, only three Te-dependent potentials have been
published.
In general, these potentials have provided us interesting insights in non-equilibrium dy- namics. For instance, the increase of the lattice parameter at high Te was responsible for
non-thermally accelerated melting (or “phase explosion”), which was found in the case of laser-excited Si [51] and Au [3]. Such change of the lattice parameter was also a contributing factor in the mechanism of laser-induced ablation in Au [87, 88]. Other notable application areas of such potentials are SHI sputtering studies [83] (see also Sec. 3.3.2.3).
2.3.3 Incorporating electrons explicitly
We briefly overview selected methods to study radiation damage that incorporate the effects of electronic excitations explicitly, rather than as an effective medium. These are based on the Ehrenfest dynamics, where atoms follow trajectories based on quantum mechanical description of electrons. Such methods include time-dependent tight-binding (with explicit electrons) [89] and real-time time-dependent density functional theory (RT TD-DFT), which includes an accurate ab initio electronic structure [90, 91].
The above models were used to provide with theoretical calculations of the electronic stop- ping power [92, 93]. The models point to a complex dependence of the Se on both the local
environment (for instance non-directionality of the drag force [94]) and the ion velocity and can give useful insights into the thus-far assumed electron-stopping power cutoff Ec [91]. Because
of the spatiotemporal scales that the RT TD-DFT model can access, the studies are performed with high-energy pka’s that are allowed to channel only (and hence no direct atomistic colli- sions occur). Tight-binding can typically access thousands of atoms and allows for low energy pka (∼ 1 keV) studies.
Both of the above modes are useful to evaluate the non-adiabatic effects on radiation damage, allowing for an energy transfer from ions to the electrons. Therefore these can capture the electronic stopping power, however cannot fully account for the electron-phonon coupling
effect, which can transfer the energy from the excited electrons to ions too. This is because Ehrenfest dynamics schemes do not take into account phonon emission as they do not treat the ions quantum-mechanically. The correlated electron ion dynamics method which allows to simulate very few atoms [95–98] could be a potential solution to evaluate and gain an understanding of the electron-phonon coupling process from ab initio.