Finite difference scheme

Alternative and similar 2T-MD schemes were used by Zhigilei and Ivanov [60], Duvenbeck [207–209], which were prototyped by Caro and Victoria [74]. We briefly discuss the approach by Zhigilei et al. who expressed the electron-phonon coupling as an additional (non-stochastic) force, which is proportional to the electron-phonon coupling strength, G, and the temperature difference between the ionic and electronic system, as follows:

mi

∂vi

∂t = Fi+ ξmivi. (4.35)

Here, Fi is the classical Newtonian force on the atom i and ξmiviis the 2T driving term related

to the electron-phonon coupling via

ξ = V G · (TP e− Ti)

j∈V mj(vj0)2

, (4.36)

where subscript j runs over all atoms in a voxel of volume V . A full derivation of this method can be found in [60]. The main advantage of this approach is its simplicity making it relatively easy to implement. However, a non-equilibrium Langevin thermostat reflects not only the statistical nature of the electron-ion scattering, but also allows for selective phonon excitations in future implementations, through changes to the random force spectrum, ˜R.

4.3.2 Finite difference scheme

The continuum electronic part of the 2T model (Eq. 2.3a) remains almost unchanged and can be cast into the following equation:

Ce

∂Te

∂t − ∇ · [κe∇Te] = G(Te− Ti) + GesT

0

i. (4.37)

Here, G is the coupling constant for the electron-phonon interaction and Ges is related to the

rate of electronic stopping. Note the added extra term (GesTi0) - this represents a source term

to balance the energy lost by the ionic system through electronic stopping with T_i0representing the temperature of the subsystem with velocities vi > vcut. We discuss the finite difference

scheme solutions to Eq. 4.37, which is of a form of a heat diffusion equation.

Robust algorithms allowing us to solve the heat equation are central to the framework de- scribed. In the continuum model, for the original iTS (two-temperature) method, two coupled heat equations need to be solved for temporal and spatial dependence, while the extended iTS

is expressed by three conservation equations, which can be recast to heat diffusion equation forms. The simplest implementation is based on a standard forward in time, centred in space, finite difference method using explicit time-stepping (also known as Euler’s method). Follow- ing [210] we define the thermal diffusivity as α ≡ _Cκ, where κ is the thermal conductivity and C is the heat capacity. We further denote the heat sink or source term as ˙q and rewrite the heat equation as follows:

∂T ∂t − α∇

2_{T =} q˙

C. (4.38)

This equation (Eq. 4.38) is closed and first order in time and second order in space and hence requires one boundary condition in time and surrounding boundary conditions in space. Based on the even space-time discretisation (equal spacing of space points xi with intervals of size

∆x = xi+1− xi and time-steps tn at intervals ∆t = ti+1− ti) we can write the derivative

approximations as: ∂T ∂t |xi,tn+1/2 ' T_in+1− Tn i ∆t (4.39a) ∂T ∂x |xi+1/2,tn ' T_i+1n − Tn i ∆x . (4.39b)

By taking the differences of the derivative approximations one can arrive at the expression for the second derivative:

∂2T ∂x2 |xi,tn' ∂T ∂x |xi+1/2,tn− ∂T ∂x |xi−1/2,tn ∆x ' T_i−1n − 2Tn i + Ti+1n (∆x)2 . (4.40)

Inserting this into the heat equation (Eq. 4.38) for 1-D conduction we obtain:

T_in+1− Tn i ∆t − α T_i−1n − 2Tn i + Ti+1n (∆x)2 = ˙ q C. (4.41)

Finally, we can solve this for the temperature at a new time-step (T_in+1), i.e. T_in+1= T_in+ ∆t  αT n i−1− 2Tin+ Ti+1n (∆x)2 + ˙ q C  . (4.42)

This is the core of the so-called forward time integration algorithm. It is an explicit time- stepping algorithm, meaning that each new temperature at n + 1 is calculated independently. Note the algorithm’s simplicity in that (ignoring the source term) the new temperature is the weighted average of the old temperature at the point Tn

i and its neighbours Ti±1n .

For the α parameter independent of any other parameters (such as temperature) it is convenient to define the Fourier mesh number, which permits further simplification of Eq. 4.42.

F = α ∆t

(∆x)2. (4.43)

The Fourier number can be thought of as the ratio of time-step size to the time required to equilibrate one space interval of size ∆x. The Fourier mesh also defines the stability of the solution. In the 1-D case for F > 1₂ leads to exponentially unstable solutions.

4.3. Two-temperature molecular dynamics

Extending the grid into 3-dimensions with j, k indices denoting y, z axes respectively, Eq. 4.42 becomes

T_i,j,kn+1 = F · (T_i−1,j,kn + T_i+1,j,kn + T_i,j−1,kn

+T_i,j+1,kn + T_i,j,k−1n + T_i,j,k+1n − 6T_i,j,kn ) + ∆tq˙

C, (4.44)

with the modified stability criterion requiring now that F ≤ 1₆. In order to design a simulation one needs to choose a mesh in space first and then choose ∆t to satisfy the stability criteria of Fourier mesh, i.e. ∆t ≤ (∆x)_6α2.

In this discussion we have assumed that α is temperature independent, thus ∇ · [α∇T ] = α∇2_{T simplification was made. This however will not be the case in general, particularly at}

phase transitions (changes in heat capacity) and for the Laplacian pre-factors found in the extended thermal spike model. The procedure for finite difference solutions of this class of problem is roughly as follows. We freeze the coefficients by considering the problem locally [211] and proceed as in the constant coefficient case. Simply put, α will be different for each of the interactions of the T_i+1,j,kn site with its neighbours and hence α will take the average value of the variable of interest between the interacting cells as the argument. An explicit finite difference scheme for the updated site T_i,j,kn+1 which results from the interaction between T_i,j,kn and T_i+1,j,kn of α = κ(T )/C(T ) strength is given by

T_i,j,kn+1 = ∆t (∆x)2 κ h 1 2(Ti+1,j,kn + Ti,j,kn ) i C(T_i,j,kn ) · (T n i+1,j,k− Ti,j,kn ) + . . . + ∆t ˙ q_i,j,kn C(T_i,j,kn ). (4.45) Use of this algorithm was employed is similar simulations in the past to simulate low-energy electronic excitation in atomic collision cascades (for instance by Duvenbeck and Wucher [207]). Similar schemes were devised in [45, 53].

Using this formulation a stability condition can be derived [211]. This condition will depend on the frozen coefficients involved and the key idea is to choose the most conservative time- step, covering all possible values of the frozen coefficient. The variable coefficient dependencies are quite complicated and hence bounds for the coefficients could only be estimated for a given time-step (at most a short time period) in the simulation. Therefore an adaptive time- stepping is the only practical solution. In 2T-MD both the ionic and the electronic systems are solved with adaptive (variable) time-steps, ∆tM D and ∆tF D, respectively, and therefore

the FD time-step is constrained to be an integer multiple of the MD time-step.