Threshold Displacement Energy - Theory and Simulation

Theory and Simulation

The threshold displacement energy is a materials property relevant during high-energy particle radiation of materials. The maximum energy that an irradiating particle can transfer in a binary collision to an atom in a material is given by (including relativistic effects)

where E is the kinetic energy and m the mass of the incoming irradiating particle and M the mass of the material atom. c is the velocity of light. If the kinetic energy E is much smaller than the mass of the irradiating particle, the equation reduces to

In order for a permanent defect to be produced from initially perfect crystal lattice, the kinetic energy that it receives must be larger than the formation energy of a Frenkel pair. However, while the Frenkel pair formation energies in crystals are typically around 5–10 eV, the average threshold displacement energies are much higher, 20–50 eV. The reason for this apparent discrepancy is that the defect formation is a complex multi-body collision process (a small collision cascade) where the atom that receives a recoil energy can also bounce back, or kick another atom back to its lattice site. Hence, even the minimum threshold displacement energy is usually clearly higher than the Frenkel pair formation energy.

Each crystal direction has in principle its own threshold displacement energy, so for a full description one should know the full threshold displacement surface for all non-equivalent crystallographic directions . Then and where the minimum and average is with respect to all angles in three dimensions.

An additional complication is that the threshold displacement energy for a given direction is not necessarily a step function, but there can be an intermediate energy region where a defect may or may not be formed depending on the random atom displacements. The one can define a lower threshold where a defect may be formed, and an upper one where it is certainly formed . The difference between these two may be surprisingly large, and whether or not this effect is taken into account may have a large effect on the average threshold displacement energy. .

It is not possible to write down a single analytical equation that would relate e.g. elastic material properties or defect formation energies to the threshold displacement energy. Hence theoretical study of the threshold displacement energy is conventionally carried out using either classical or quantum mechanical molecular dynamics computer simulations. Although an analytical description of the displacement is not possible, the "sudden approximation" gives fairly good approximations of the threshold displacement energies at least in covalent materials and low-index crystal directions

An example molecular dynamics simulation of a threshold displacement event is available in . The animation shows how a defect (Frenkel pair, i.e. an interstitial and vacancy) is formed in silicon when a lattice atom is given a recoil energy of 20 eV in the 100 direction. The data for the animation was obtained from density functional theory molecular dynamics computer simulations.

Such simulations have given significant qualitative insights into the threshold displacement energy, but the quantitative results should be viewed with caution. The classical interatomic potentials are usually fit only to equilibrium properties, and hence their predictive capability may be limited. Even in the most studied materials such as Si and Fe, there are variations of more than a factor of two in the predicted threshold displacement energies. The quantum mechanical simulations based on density functional theory (DFT) are likely to be much more accurate, but very few comparative studies of different DFT methods on this issue have yet been carried out to assess their quantitative reliability.