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Integration of electronic eﬀects into molecular dynamics simulations of collision

cascades in silicon from ﬁrst-principles calculations

Thomas Jarrin and Nicolas Richard∗

CEA, DAM, DIF, F-91297, Arpajon, France

Johannes Teunissen and Fabiana Da Pieve

Royal Belgian Institute for Space Aeronomy BIRA-IASB, 1180, Brussels, Belgium

Anne H´emeryck†

LAAS-CNRS, Universit´e de Toulouse, CNRS, Toulouse, France

The inclusion of sophisticated density-dependent electronic stopping and electron-phonon coupling

calculated with ﬁrst-principles methods into molecular dynamics simulations of collision cascades

has recently become possible thanks to the development of the so-called EPH (for Electron-PHonon)

model. This work aims at employing the EPH model in molecular dynamics simulations of collision

cascades in Si. In this context, the electronic stopping power is investigated in Si at low energies with

Ehrenfest Dynamics calculations. Also, the parametrization of the EPH model for Si, from ﬁrst-

principles Ehrenfest Dynamics simulations to actual molecular dynamics simulations of collision

cascades, is performed and detailed. We demonstrate that the EPH model is able to reproduce

very closely the density-dependent features of the energy lost to electrons obtained with ab initio

calculations. Molecular dynamics collision cascade simulations results obtained in Si using the EPH

model and the simpler but widely employed Two Temperature Model (TTM) are compared, showing

important discrepancies in the collision cascades results obtained depending on the model employed.

I. INTRODUCTION

Accurate Molecular Dynamics (MD) simulations of

collision cascades into semiconducting materials are of

high interest for applications in hazardous radiation en-

vironments such as space [1] and nuclear power facilities

[2], where sensitive microelectronic and optoelectronic

devices like bipolar junction transistors and image sen-

sors are subjected to intense ﬂuxes of energetic parti-

cles. Neglecting the non-Coulomb interactions occurring

at very high energies only (on the order of MeV), ener-

getic charged particles into matter transmit their kinetic

energy through nuclear stopping and electronic stopping:

when the energies at stake are very high, electronic stop-

ping is by far the leading loss mechanism, whereas when

the energies are low, nuclear stopping becomes the dom-

inant mechanism. In a collision cascade event, defects

are created as a result of Coulomb interactions between

the atoms set in motion and those at rest, i.e. nuclear

stopping. Electronic stopping, which describes the exci-

tation of electrons of the target material by the projectile,

is not directly responsible for the creation of defects in

collision cascades. However, an accurate evaluation of

the energy going into atomic displacements, and thus a

complete model of collision cascades, must take into ac-

count the energy lost to electrons. Indeed, the latter can

have important implications on the dynamics of the ions

and the recombination of the defects. The well-known

Lindhard-Scharﬀ-Schiott (LSS) theory is an example of

∗nicolas.richard@cea.fr

†anne.hemeryck@laas.fr

successful attempts to develop a uniﬁed model for elec-

tronic and nuclear stopping [3,4]. Nevertheless, this the-

ory does not allow for a dynamic atomic-scale treatment

of collision cascades. Thus, for many years, MD simula-

tions could not take into account electronic stopping but

only nuclear stopping via repulsive empirical potentials,

the best-known being the Moliere potentials [5] and the

ZBL potentials [6]. Fortunately, in the last two decades,

several models accounting for electronic stopping and/or

electronic-phonon coupling have been developed to be

used in addition to MD simulations, which are inherently

able to correctly account for nuclear stopping through the

interatomic potential.

The simplest form of these models accounts for elec-

tronic stopping, via a friction force, but does not account

for electron-phonon coupling [7]. Another form, probably

the most employed one, is the derivation of the Two Tem-

perature Model (TTM), developed by Duﬀy and Ruther-

ford [8] for MD simulations from earlier works by Caro

and Victoria [9]. This model accounts for both electron-

phonon coupling and electronic stopping in the frame-

work of a Langevin thermostat [10]: a friction force acts

as the electronic stopping and a stochastic force term em-

bodies the electron-phonon coupling. Despite the major

breakthrough that this model represented, it suﬀers from

certain limitations. Firstly, the electronic density is con-

sidered constant in the entire model The electronic stop-

ping power thus cannot be crystal direction-dependent

as it was conﬁrmed to be by recent experiments [11] and

Time Dependent Density Functional Theory (TDDFT)

calculations [12]. Secondly, a recent paper pointed out

[13] the large uncertainties in the choice of values for

some parameters of the model, due to the lack of the-

2

oretical or experimental data in the literature. In re-

sponse to the limitations raised above, a model, named

EPH (for Electron-PHonon) [14,15], has been developed

recently, taking as a basis the framework of a Langevin

thermostat, such as the TTM, but incorporating the no-

tion of spatial locality in the model through the possi-

bility to deﬁne a non constant electronic density in the

simulation box [14]. In the EPH model, the electron-

phonon coupling is seen as an electronic stopping pro-

cess, which allows the construction of a uniﬁed model for

ion-electron interactions. Since the electronic density is

not constant in the model, values of the electronic stop-

ping power with respect to the electronic density must

be provided as input to the simulation. To obtain it, it

is necessary to resort to Ehrenfest Dynamics (ED) calcu-

lations of electronic stopping power. The ED framework

can go beyond linear response and Real-Time TDDFT

(RT-TDDFT) by propagating both the dynamics of the

electrons with RT-TDDFT and the dynamics of the nu-

clei via integration of the classical equations of motions

of the nuclei. In the following, the acronym TDDFT is

used to refer to ED calculations as it is commonly done

in the literature of electronic stopping calculations. As

TDDFT has been shown to correctly describe the high

energy electronic stopping power regime [16], but also

electron-phonon processes occurring at very low energies

(typically meV) [17], the EPH model therefore provides a

framework for MD simulations accounting for electronic

eﬀects with very high accuracy. The model and its use

on Ni and Ni alloys is described in [15] and [18].

The purpose of the present paper is to parametrize

the EPH model for a semiconductor, silicon Si, with

TDDFT calculations. We aim to demonstrate its ability

to reproduce electronic stopping calculated by TDDFT

within MD simulations and we compare the results of

MD collision cascades in Si with diﬀerent EPH and TTM

parametrizations. Section II describes the EPH model

and gives the computational details of our TDDFT and

MD simulations. Section III is dedicated to TDDFT cal-

culations of electronic stopping in Si for diﬀerent energies

and crystal directions. In Section IV is detailed how to

parametrize the EPH model. The results of the collision

cascades in Si obtained with the EPH model and the

TTM are compared and discussed in Section V.

II. METHODS

In this section, the EPH model is described and the

methodologies as well as computational details employed

in TDDFT and MD simulations are given.

A. EPH model

The EPH model is currently implemented as a ﬁx

plug-in for LAMMPS [19]. It was developed and im-

plemented by A. Tamm and A.A. Correa. It is based

on the Langevin framework they derived in [14] where a

system of ions exchanges energy with a bath of electrons

of temperature Te. In this framework, the scalar values

of friction and random forces of usual Langevin models

(among them the TTM), acting as the electronic stop-

ping and the electron-phonon coupling in the TTM re-

spectively, are replaced by many-body tensor notations,

allowing an accurate description of electronic stopping

and electron-phonon coupling processes under the same

model. The usual MD equation of motion for particle I

therefore becomes:

mI~aI=~

Fadiab

I−X

J

BIJ ~vJ+X

J

WIJ ~

ξJ(1)

In the above equation, mIis the mass of particle I,~aI

its acceleration, ~

Fadiab

Ithe conservative forces (deriving

from the interatomic potential in MD simulations), ~vIits

velocity, PJBIJ ~vJthe friction force acting on particle I

where BIJ are IJ components of the Btensor deﬁning

the magnitude of the friction force and PJWIJ ~

ξJthe

random force acting on particle Iwhere ~

ξJare uncor-

related Gaussian random variables normalized to 2kBTe

and WIJ are IJ components of the Wtensor deﬁning the

magnitude of electron-phonon coupling. The Wand B

tensors are related via the following expression deriving

from the ﬂuctuation-dissipation theorem [20]:

BIJ =X

K

WIK WT

JK (2)

The magnitudes of both the electronic stopping power

and the electron-phonon coupling entirely lie within the

WIJ terms. This choice made in the EPH model is based

on the works of Caro et al. [9] and Koponen [21] from the

early 90s, as well as on more recent results by Caro et al.

[17], which assert that electronic stopping and electron-

phonon coupling are the “opposite limits of the same ba-

sic governing equations” (or mechanism) [21]. In other

words, based on the same physics of energy exchange,

the electronic stopping power describes the case of high

velocity projectiles sampling regions of various electronic

densities, while electron-phonon coupling corresponds to

low velocity atoms oscillating around their equilibrium

position of relatively low electronic densities. This aspect

is made explicit by the correlations between friction and

random forces shown in (2). The deﬁnition of Wmust

ensure the conservation of angular and linear momentum

[14] in order to have a rich electron-phonon mode. To this

end, the matrix terms WIJ of the Wtensor are deﬁned

as follows in the MD-EPH model:

WIJ =

−αJ( ¯ρJ)ρI(rIJ )

¯ρJ

~eI J ⊗~eI J (I6=J)

αI( ¯ρI)X

K6=I

ρK(rIK )

¯ρI

~eI K ⊗~eI K (I=J)

(3)

3

with ρI(rIJ ) the electronic density created by atom

Iat a distance rIJ , ¯ρI=PJ6=IρJ(rIJ ), ~eIJ are unit

vectors of the IJ direction and αI( ¯ρI) parameters which

control the friction and random forces magnitudes. This

last term is the key parameter to be speciﬁed by the user.

From the user’s point of view, in the EPH model, elec-

tronic stopping power and electron-phonon coupling are

deﬁned by the same single α(ρ) parameter. In theory,

the low ρpart of α(ρ) coupled to the low velocities of

the ions governs the electron-phonon coupling part. This

is very diﬀerent from the strategy adopted in the TTM

framework, in which two distinct parameters must be de-

ﬁned for electronic stopping power and electron-phonon

coupling. The EPH model was developed to reproduce

TDDFT data on electronic stopping. Thus, the αparam-

eters appearing in (3) must be optimized to ﬁt TDDFT

results. This aspect is the most diﬃcult and most im-

portant part of parametrizing the EPH model, and this

article describes how to do it for Si in Section IV. The

whole TDDFT calculations and the process of ﬁtting to

TDDFT were already performed on Ni and Ni alloys in

[15] and [18].

In the model, the electronic density is not the real one

because it would require solving the Schr¨odinger equa-

tion at each timestep of the MD simulation. The elec-

tronic density is approximated by considering that the

electronic density around each atom of the system is that

of the atomic sphere density of the isolated atom. At a

given point of the simulation box, the electronic density

is thus the sum of the contributions of the atomic sphere

electronic density of all the atoms located within a certain

cutoﬀ radius of this given location. The users of the EPH

model must provide the densities of the atomic spheres of

the species involved. This atomic sphere approximation

is known to work well for metals [22] for which electrons

are delocalized, but the more localized covalent bonds of

semiconducting materials might decrease the accuracy of

this approximation. We discuss this aspect in Section IV.

In this work, the OPIUM code [23] is used to calculate

the atomic sphere density of Si.

B. TDDFT calculations

TDDFT calculations of electronic stopping are per-

formed using the QB@LL code [24,25], following the pro-

cedure detailed in [26]. All our simulations are performed

on 3×3×3 bulk diamond supercells of Si with the exper-

imental lattice constant of 5.431 ˚

A [27]. Periodic bound-

ary conditions are employed to accurately model a bulk

system. Calculations are carried out with the PBE ap-

proximation for the exchange and correlation functional

and a norm conserving pseudopotential with 4 valence

electrons (3s23p2) is used. The results obtained with

this pseudopotential are compared in Section III with re-

sults we obtained with another pseudopotential with 12

valence electrons. This 12 valence electrons pseudopo-

tential is adapted to work with PBE functionals from

the one generated by Lee et al. in [12] with the OPIUM

code [23]. Indeed, it has been vividly demonstrated in

[16] and [28] that, at least at high energies, the core elec-

trons of the target and of the projectile can be excited.

We use a plane-wave cutoﬀ of 100 Ry with the 4 valence

electrons pseudopotential and a cutoﬀ of 220 Ry with the

12 valence electrons pseudopotential.

The initial condition required for the real-time prop-

agation is obtained by ﬁrst performing a self-consistent

calculation of the system consisting of the supercell and

the projectile. Then, the electronic wavefunctions are

propagated in time from the wavefunctions obtained from

the static calculation, and the classical equations of mo-

tions of the nuclei are integrated in time, with force calcu-

lations derived from the electronic Hamiltonian and the

electronic wavefunctions.

The velocity of the projectile is kept constant, and the

forces on all atoms are set to zero. Thus, we only record,

through the increase of the total energy, the contribu-

tion of electronic excitations. In this paper, we are only

interested in neutral Si projectiles in bulk Si diamond

at energies ranging from 10 keV to 200 keV. These en-

ergies are typical of collision cascades events. Our goal

is to have a statistical treatment of MD simulations of

collision cascades, but with energies higher than a few

tens of keV, this is not possible because of the associated

computational costs.

Below 50 keV, we use a timestep of 1 as, and from

50 keV and above we employ a timestep of 0.5 as. The

timesteps are chosen to ensure energy conservation, no

inﬂuence on the calculations of the electronic stopping

powers and reasonable computational cost.

We simulated the ions and electrons dynamics in the

trajectories of <001>,<110>and <111>center and oﬀ-

center channels as well as in directions incommensurate

with the crystal lattice. Incommensurate directions are

chosen following the indications of [26], such that they

avoid channeling directions and head-on collisions. All

the directions cited above were used for the optimiza-

tion of the α(ρ) function of the EPH model, but elec-

tronic stopping powers were only calculated on trajecto-

ries in the <001>,<110>and <111>directions. To

extract the stopping power from the simulations in the

channels, we calculated the slope of the linear regres-

sion of the total energy against the distance xtraveled

by the projectile, but only between x= 1/2×alat and

x= 5/2×alat for the <001>direction, x=√2/2×alat

and x= 5√2/2×alat for the <110>direction, and

x=√3/2×alat and x= 5√3/2×alat for the <111>di-

rection. Doing this, we ignore the short transient state at

the beginning, and we completely eliminate the contribu-

tion of the lattice without having to resort to Born Op-

penheimer Approximation-MD (BOA-MD) simulations.

See [29] for more details on this.

4

C. MD simulations

We use the LAMMPS code for MD simulations [30].

The cascades are initiated with 10 keV PKAs (Primary

Knock-on Atoms) in Si. Boxes of 1 000 000 atoms made

of 50×50×50 diamond-like unit cells and 4 096 000 atoms

made of 80×80×80 diamond-like unit cells are employed.

The simulation boxes are divided into two areas: in the

outer cells the velocities are rescaled to maintain the

temperature at a desired value (thus absorb the ther-

mal wave), and the inner cells form an NVE ensemble

in which the atoms evolve freely to simulate the collision

cascade.

The initialization of the simulation is done by scaling

the velocities of all atoms, such that the overall temper-

ature is the desired one (300 K in the current work).

For calculations carried out with the TTM, the system is

equilibrated for at least 20 ps with a timestep of 1 fs.

With the EPH, we found that this equilibration time

needed to be increased to 100 ps. Since the velocity of

the atoms drastically changes throughout the cascade,

the integration timestep is changed during the simula-

tions. We imposed the condition that no atom moves

more than 0.02 ˚

A between two steps of the simulation,

with a timestep varying between 0.001 fs and 1 fs.

We employed the Stillinger-Weber (SW) potential de-

veloped in [31]. To better describe short interatomic dis-

tances, the SW potential is combined to a repulsive two-

body potential. The chosen repulsive potential is the

Ziegler Biersack Littmark potential (ZBL) [6]. SW and

ZBL potentials are combined together through a Fermi

function as in [32]:

Vtot(r) = (1 −F(r))VZBL(r) + F(r)VSW (r) (4)

where ris the distance between two atoms, Vtot is the

total potential, VZB L the repulsive ZBL potential, VSW

the SW potential and Fthe Fermi function used to link

the two potentials. The expression of the Fermi function

as well as its parameters values used for Si can be found

in [33].

To correctly account for the stochasticity of the cas-

cades, the distributions and statistical quantities are

based on 75 simulations in distinct and independent di-

rections [34]. We usually use sets of 100 simulations but

the high computational cost of simulations with the EPH

model compelled us to reduce this number to the already

satisfactory one of 75. The method employed to choose

the directions is the “Symmetry” method detailed in [34].

Depending on the simulations performed, the elec-

tronic stopping TTM parameter (γs) was optimized to

best reproduce the TDDFT stopping data, or was de-

rived from SRIM calculations [35]. Details of the TTM

and EPH parameters employed are given when needed.

III. ELECTRONIC STOPPING POWER

CALCULATIONS

In this section, we are interested in extracting the elec-

tronic stopping power from the evolution of the total en-

ergy with respect to the distance traveled by the pro-

jectile in TDDFT calculations, in center and oﬀ-center

(1/2) <001>channel trajectories for energies ranging

from 10 keV to 200 keV. By oﬀ-center (1/2) we mean

that the initial position of the projectile is (1.125 ×alat,

1.5×alat, 0.0). Trajectories along the <110>and <111>

channels have only been investigated with 100 keV pro-

jectiles.

The purpose of our TDDFT calculations is to allow

parametrizing the EPH model for MD simulations of col-

lision cascades. Thus, we are interested in low-energy

projectiles, which explains why simulations are carried

out with Si projectiles from 10 keV to 200 keV. While for

higher energies it has been proven to be mandatory, at

least in Ni [16], to employ pseudopotentials with semicore

electrons in the valence, no pseudopotential-dedicated

study was performed for the low energies of interest for

us. However, a detailed study with a 12 valence electrons

pseudopotential has been performed for Si in [12].

Here, we ran some simulations to determine whether it

is necessary to employ a pseudopotential with semicore

electrons in the valence, which would have a consider-

ably higher computational cost. For comparison, we per-

formed a simulation in the <001>center channel with

a projectile of 700 keV with both a 4 valence electrons

pseudopotential and a 12 valence electrons pseudopoten-

tial. The pseudopotential with 4 valence electrons gives

a stopping power of 33.9 eV/˚

A, compared to 53.6 eV/˚

A

with the pseudopotential with 12 valence electrons. The

reported value with the pseudopotential with semicore

electrons in the valence is in good agreement with the

calculations performed with a similar pseudopotential in

[12]. Making the same comparison for a 200 keV pro-

jectile along the same trajectory, we ﬁnd an electronic

stopping of 16.7 eV/˚

A with the 4 valence electrons pseu-

dopotential and of 16.0 eV/˚

A with the 12 valence elec-

trons pseudopotential.

The lower value found with the pseudopotential explic-

itly integrating semicore electrons is counter-intuitive. It

can be explained by the fact that the 12 valence elec-

trons pseudopotential generated with OPIUM supports

only one projector per angular moment, which means

that it had to be generated for the reference conﬁgu-

ration 2s22p63s03p0and not the neutral conﬁguration

2s22p63s23p2. As a result, it suﬀers from transferability

problems which could easily explain uncertainties in the

electronic stopping calculations. However, the results are

similar enough to state that below 200 keV a minima, a

pseudopotential with more than 4 valence electrons is of

no use. All the results presented in the following were

obtained with the 4 valence electrons pseudopotential.

In Fig. 1, the electronic stopping powers of initially

neutral Si projectiles into bulk Si along the center <001>

5

channel and the oﬀ-center <001>channel (1/2) are

shown with respect to the velocity of the projectile. The

results of TDDFT calculations from [36], experimental

results [11] and SRIM results are also shown for compar-

ison.

FIG. 1. Electronic stopping powers with respect to the ve-

locity of the projectile in a.u. calculated for projectiles of

energies ranging from 10 keV to 200 keV (solid markers) and

compared to stoppings from other computational (empty di-

amond markers) and experimental (empty triangle markers)

references and calculated with SRIM (dashed line).

According to Fig. 1, the increase in electronic stopping

power in Si for projectiles between 10 keV and 200 keV

in the <001>channel is clearly linear with the projec-

tile velocity. Our calculations compare really well with

the results presented in [36]. A slight discrepancy can

be noticed, which could not be explained neither by the

use of LDA vs PBE, nor by the diﬀerent lattice param-

eters employed, according to additional calculations we

performed.

The stopping powers found for the oﬀ-center <001>

trajectories are a bit higher than those for the center,

and this discrepancy increases with the projectile ve-

locity. Our results for the center channel and those of

[36] are in good agreement with the state-of-the-art ex-

perimental results of [11] for the <001>center channel.

This gives us conﬁdence in the reliability of the electronic

stopping powers calculated from TDDFT despite the fact

that our stoppings are 2 to 3 times smaller than SRIM

stopping powers. This last point is of course partly due

to the fact that we only consider results for the <001>

channels, whereas the electronic stopping in SRIM is a

kind of average over all possible directions. Some of the

stoppings reported for the diﬀerent sampled trajectories

for the 100 keV projectile in Table Iare already closer to

TABLE I. Electronic stopping powers calculated with

TDDFT for various directions for a neutral Si projectile of

100 keV.

Elec. stop. (eV/˚

A)

<001>center channel 11.1

<110>center channel 9.8

<111>center channel 13.8

<001>oﬀ-center channel (1/2) 14.2

<001>oﬀ-center channel (1/4) 19.5

<110>oﬀ-center channel 12.2

SRIM 35.8

the SRIM value.

The disagreement between SRIM and TDDFT stop-

pings for Si in Si at low energies has already been pointed

out in [36]. More recent work by Lee et al. at higher en-

ergies in [12] reports TDDFT electronic stoppings for ini-

tially charged Si+12 projectiles up to twice greater than

the SRIM stoppings, compared to TDDFT stoppings up

to twice lower for neutral Si+0 projectiles, which is con-

sistent with our results with neutral Si projectiles.

Trying to reproduce SRIM values of electronic stopping

power requires running very long TDDFT simulations on

various random trajectories (see [29]) and with various

initial projectile charge states to obtain converged values

of the stopping. Doing so is beyond the scope of this

paper.

IV. PARAMETRIZATION OF THE EPH

MODEL

This section is dedicated to the parametrization of the

parameter α(ρ) of the EPH model. Following (3), αis

expressed in (eV.ps/˚

A2)1/2(default units of the model).

The parameter that we really have to adjust, according

to [15], is homogeneous to (eV.ps/˚

A2), thus the square of

α. It is sometimes called βin [15] and [18], has the same

role in BIJ as αhas in WI J , as can be understood from

(3) and (2), and can be seen as the electronic stopping

power of the projectile divided by its velocity. The origi-

nal papers of the EPH model [14,15,18] do not explicitly

mention βin the equations although they actually name

it in the text and optimize it. Although we actually do

optimize the βparameter and not the αparameter, as α

appears in the equations, the αparameter is the param-

eter discussed in the following.

For any set of distinct projectile trajectories, the best

parametrization is the one that gives an evolution of the

energy lost to the electrons as a function of the distance

that is the closest to that obtained with TDDFT calcu-

lations. The energy lost to the electrons is a direct out-

put of the EPH model, whereas it requires an additional

step to be obtained from TDDFT calculations. To ob-

tain only the energy transmitted to the electrons by the

projectile ion, the contribution of the lattice to the total

energy must be subtracted. In practice, the energy with

6

Run RT-TDDFT and BOA-MD

simulations on various trajectories

Eelec(r) = ETDDFT(r) - EBOA(r)

Provide initial guess for α(ρ)

Launch EPH-MD simulations on the

same trajectories and in the same

conditions as in the ﬁrst step

Calculate distance D between two

sets of runs

D < cutoﬀ ?

Right α(ρ) is found

Yes

No

Update α(ρ)

FIG. 2. Optimization scheme of the α(ρ) parameter.

respect to distance obtained with a BOA-MD run on a

given trajectory EBOA must be subtracted from the en-

ergy with respect to distance obtained with TDDFT on

the same trajectory ET D DF T . At a given distance ron

the projectile path, the energy Eelec (r) lost to electrons

in a TDDFT run thus is:

Eelec(r) = ET D DF T (r)−EBO A(r) (5)

In this article, when an energy loss to electrons ob-

tained with TDDFT is mentioned, it was calculated with

(5). It doubles the number of simulations to be per-

formed because for a single trajectory, one TDDFT sim-

ulation and one BOA-MD simulation are required. Since

the parameterization of the EPH model requires to run

many TDDFT calculations in diﬀerent trajectories, it is

computationally prohibitive to ﬁt the model with simula-

tions launched with too low velocities. Indeed, the lower

the initial energy (velocity) of the projectile, the longer

the simulation time. Thus, we ﬁt the EPH model with

TDDFT calculations initiated with 100 keV projectiles,

even if we will then perform our MD simulations of col-

lision cascades with lower initial energies. By doing so,

we reach an aﬀordable computational cost. This approx-

imation is reasonable considering that the EPH model

assumes that the electronic stopping is linear with ve-

locity, which is always the case for metals, for which the

EPH model has been developed. In the case of Si, Lim

et al. demonstrated in [36] that the electronic stopping

is metal-like above 3 keV, which means that it is linear

with the velocity.

Our calculations in the <001>channel conﬁrmed this

observation (see Fig. 1) for projectiles from 10 keV to

200 keV . Also according to the work of Lim et al., below

3 keV and above 60 eV, the bandgap induces changes as

electronic stopping power still increases with velocity but

faster than above 3 keV. Below 60 eV, the stopping was

found to be non-zero. However, at low energies (veloci-

ties) the contribution of electronic eﬀects is less impor-

tant than at high energies and the initial energies we use

for MD simulations of collision cascades are well above

3 keV. Consequently, the errors coming from this linear

stopping approximation should be small.

To rank the diﬀerent MD-EPH (MD-TTM)

parametrizations, an appropriate distance measure

between the energy lost to electrons along a given

projectile trajectory during the reference TDDFT simu-

lations and the MD-EPH (MD-TTM) simulations must

be deﬁned. We use the Mean Absolute Error (MAE):

MAE =1

n

n

X

i|EMD (ri)−ET D DF T (ri)|(6)

where nis the number of points sampled along the tra-

jectory and riis the distance traveled by the projectile

associated to point i.EM D in this equation describes

either the energy lost to electrons during a MD-EPH or

a MD-TTM simulation.

An optimization scheme for the α(ρ) parameter of the

EPH model can now be established. This scheme is pre-

sented in Fig. 2.

We ﬁt the EPH model using ten diﬀerent trajecto-

ries: <001>center channel and two oﬀ-center trajecto-

ries, <110>center channel and one oﬀ-center trajectory,

<111>center channel and also four incommensurate di-

rections, one integrating a vacancy (one missing atom)

on the projectile path. Table II summarizes the charac-

teristics of all sampled directions.

The conditions of the MD-EPH runs are the same as

for the TDDFT runs: the forces on all atoms are set

to zero and the projectile moves with a constant veloc-

ity. For comparison, MD-TTM simulations in the same

conditions are also run. In the TTM and EPH runs

in this section, the electron-phonon coupling is switched

oﬀ, so we only consider the electronic stopping. In the

TTM, the total friction coeﬃcient is actually the sum of

the electronic stopping parameter γsand the electron-

phonon coupling parameter γp[8]. This means, in the

TTM calculations, we neglect the energy exchanges be-

tween ions and electrons via the stochastic force term

(electron-phonon coupling) and we only consider the pa-

rameter γsin the friction term. Activating the electron-

phonon coupling would not change the results, but would

make necessary an extra smoothing step of the energy

lost to electrons curves.

It was found in [37] that the electronic stopping power

is a multivalued function of the electronic density: for

a single value of the electronic density there are several

values of electronic stopping. The parameters α(ρ) can-

not be a multivalued function of the density. Thus, the

stopping retrieved from the EPH model can only approx-

imate the reference TDDFT stopping. We want to make

7

TABLE II. Main characteristics of the trajectories sampled for ﬁtting the EPH model.

Initial projectile position Velocity unit vector

Center channel <001>(1.25 ×alat, 1.5×alat, 0.0) (0.0, 0.0, 1.0)

Center channel <110>(0.0, 0.0, 1.625 ×alat ) (1/√2, 1/√2, 0.0)

Center channel <111>(1.25 ×alat, 1.25 ×alat, 0.0) (1/√3, 1/√3, 1/√3)

Oﬀ-center channel <001>(1/2) (1.125 ×alat, 1.5×alat, 0.0) (0.0, 0.0, 1.0)

Oﬀ-center channel <001>(1/4) (1.0625 ×alat, 1.5×alat, 0.0) (0.0, 0.0, 1.0)

Oﬀ-center channel <110>(0.0, 0.0, 1.81 ×alat ) (1/√2, 1/√2, 0.0)

Incommensurate 1 1.25 ×alat, 1.5×alat , 0.0) (0.3620, 0.0.1134, 0.9252)

Incommensurate 1 + vacancy (1.25 ×alat, 1.5×alat , 0.0) (0.3620, 0.0.1134, 0.9252)

Incommensurate 2 (1.25 ×alat, 1.5×alat , 0.0) (0.4017, 0.1867, 0.8965)

Incommensurate 3 (1.25 ×alat, 1.25 ×alat , 0.0) (0.1922, 0.4364, 0.8790)

FIG. 3. Constant and quadratic optimized β(ρ) = α(ρ)2func-

tions.

it as true as possible to the TDDFT data using a single-

valued function for the parameter α. In [15], Caro et al.

used a complex optimization algorithm procedure with

spline and knot points for α(ρ).

In order to demonstrate the accessibility of the EPH

model, we chose to employ two simple analytical expres-

sions for α(ρ): one where αis kept constant, and an-

other one where α(ρ) has a quadratic expression of the

form (aρ +bρ2)×(1 + ec(ρ−ρf))−1. The exponential de-

cay term in the quadratic expression must be added to

the quadratic expression otherwise the stopping would

be drastically overestimated at high ρ. This is also moti-

vated by the observations made in [15] that the electronic

stopping tends to saturate for very high values of the elec-

tronic density. The quadratic expression has no constant

term in order to get it to be equal to 0 at ρ= 0.

In the following, the data corresponding to the op-

timized αparameters are shown. The best optimiza-

tion is deﬁned when the total MAE, calculated from the

sampled positions of all the studied trajectories, is the

smallest. Fig. 3represents the optimized α(ρ) param-

eters with respect to density in those two cases. The

optimized expressions of the αparameter in the constant

and quadratic cases respectively give:

•α(ρ)2=0.0037 eV/ps/˚

A2

•α(ρ)2= (0.041×ρ+0.0×ρ2)×1

1+e10(ρ−0.3) eV/ps/˚

A2

Starting from a quadratic expression of α(ρ), we obtained

a linear one in the end with the bparameter strictly equal

to 0. As the ﬁnal linear expression we obtain originates

from the optimization of an initially quadratic expres-

sion, the term quadratic will be kept in the following.

Fig. 4and Fig. 5show the energy lost to the electrons

in channeling and incommensurate trajectories obtained

with TDDFT, the EPH model and the TTM, respec-

tively. Table III displays the MAE in the constant and

quadratic EPH scenarios and the TTM ﬁtted to TDDFT

case only, for all the studied trajectories. Note that the

TTM electronic stopping parameter in the TTM curves

below named “TTM ﬁtted” (see Fig. 4and Fig. 5) was

also optimized to minimize their MAE with respect to

the TDDFT data. Contrarily, the TTM parameters of

the simulations whose corresponding curves are entitled

“TTM SRIM” are not ﬁtted to TDDFT data. In this

case, the electronic stopping power parameter γsis ob-

tained from the SRIM calculations of electronic stopping

power.

The most blatant feature, common to all the graphs

of Fig. 4, is the fact that the energy lost to electrons

with the TTM in its SRIM-derived parametrization is

far greater than the energy lost to electrons calculated

with TDDFT. Considering the existing discrepancy we

highlighted in Section III between our TDDFT electronic

stopping results and the SRIM stopping calculations, this

result was expected.

We also clearly see an increase in the slope of the en-

ergy lost to electrons when regions of higher densities are

sampled. This is particularly clear in Fig. 4(e) and all

the incommensurate directions of Fig. 5, in which the

projectile encounters high values of electronic density on

its path. However, at high values of electronic density,

the slope of the energy lost to electrons stops increasing

and stabilizes. The increase in electronic stopping thus

seems to saturate when high values of electronic den-

sity are reached. Visually, from the ﬁgures just cited,

8

(a) (b)

(c) (d)

(e) (f)

FIG. 4. Energy lost to the electrons by the projectile for all the channel trajectories studied with 100 keV Si projectiles in Si

calculated with TDDFT, the EPH model and the TTM (solid lines). The electronic density seen by the projectile in the EPH

model is represented in each case with a dashed black line. We deﬁne the zero energy reference for all the curves at the x-axis

point at which we start the ﬁtting. The curves are plotted on the same symmetric portions of the trajectories as the ones we

deﬁned in Section II B for the calculations of stopping powers. Beware that the right y-axis for the electronic density are in

log-scale, and that the x-axis are not the same in each graph.

9

(a) (b)

(c) (d)

FIG. 5. Energy lost to the electrons by the projectile for all the incommensurate trajectories studied with 100 keV Si projectiles

in Si calculated with TDDFT, the EPH model and the TTM (solid lines). The electronic density seen by the projectile in the

EPH model is represented in each case with a dashed black line. We deﬁne the zero energy reference for all the curves at the

x-axis point at which we start the ﬁtting. Beware that the right y-axis for the electronic density is in log-scale.

the EPH model also provides an increase in the stopping

power when the electronic density increases, just as in

the TDDFT results. This can be observed in Fig. 4(d),

Fig. 4(e), Fig. 4(f) as well as in all the graphs of Fig. 5, in

which the changes in the slopes of the energy lost to elec-

trons with changes in the electronic density are obvious.

This constitutes a ﬁrst qualitative satisfactory aspect of

the EPH model.

We now focus on the channel trajectories of Fig. 4.

The TDDFT trajectories of the <001>oﬀ-center chan-

nel (1/2), <110>oﬀ-center channel and <111>center

channel of Fig. 4are very well reproduced by the con-

stant parametrization of the EPH model, the density de-

pendence of the energy lost to electrons being followed

very closely by MD-EPH simulations. This is conﬁrmed

by Table III and the low MAE associated to these tra-

jectories. The quadratic parametrization also reproduces

quite well the <110>oﬀ-center channel and <111>cen-

ter channel, but does not perform as good as the con-

stant case for the <001>oﬀ-center channel (1/2): we

get a MAE of 4.2 eV for the constant case compared to

13.4 eV in the quadratic case.

The <001>center channel and <001>oﬀ-center chan-

nel (1/4) do not follow the reference TDDFT data curves

as closely as the previously cited trajectories but still are

satisfactory. The constant αfunction provides better re-

sults for the <001>center channel (MAE of 9.4 eV com-

pared to 12.9 eV), but the situation is reversed for the

<001>oﬀ-center channel (1/4) (MAE of 15.5 eV com-

pared to 8.0 eV).

10

Following the good results obtained with the EPH

model for the ﬁve trajectories already mentioned, it is

surprising that for the <110>center channel trajectory,

the MD-EPH simulations are far below the reference

TDDFT data (Fig. 4(b)). The origin of this large dis-

crepancy might be due to the fact that the electronic

density approximated by the EPH model for this trajec-

tory might be very low compared to the actual electronic

density that the projectile sees in the TDDFT simulation.

However, we compared the EPH and ab initio electronic

densities calculated with Quantum Espresso [38] for var-

ious trajectories including the center <110>. We found

that in all cases, the densities are similar. Therefore, it

cannot explain the discrepancies we observe for the center

<110>channel between EPH and TDDFT.

Choosing a diﬀerent analytic expression α(ρ) would

certainly help to obtain a better agreement for the case

of the <110>channel, but it could also mean working

with more complex and possibly physically meaningless

expressions. A possibility, which is not to be omitted

to explain the observed discrepancies, is that the <110>

center channel emphasizes a non-linear aspect of elec-

tronic stopping the EPH model is unable to capture.

With the EPH model, at low electronic densities, the

electronic stopping power is roughly proportional to the

electronic density. Moving from Fig. 4(b) to Fig. 4(a), we

see that the electronic density is multiplied by a factor of

about 4-5, and that the same is observed for the energy

lost to electrons with the EPH model. However, the en-

ergy lost to electrons obtained with TDDFT is nearly un-

changed between the <001>center channel and <110>

center channel (in agreement with the results of Table I),

whereas we observed that the ab initio electronic density

is very close to that displayed in Fig. 4(a) and Fig. 4(b).

Further work should be conducted to be conclusive on

this point.

In Fig. 4(e) one can notice a step-like behavior of the

energy lost to electrons in the EPH model. This behavior

is intrinsic to the model, but arises for lower values of the

electronic density in the quadratic parametrization of the

EPH model. Indeed, the parameter α(ρ) in the quadratic

parametrization smoothly decreases to zero. It means

that when ρexceeds the value for which α(ρ) = 0, no

energy is transferred between ions and electrons and that

the energy lost to electrons levels oﬀ until the projectile

reaches a lower value of ρ.

Although the TTM ﬁtted energy loss curves in the

channel cases of Fig. 4are linear, they do not per-

form that bad in many cases, as can be observed in

Fig. 4. According to Table III, the MAE is even smaller

in the TTM ﬁtted case than in both MD-EPH scenar-

ios for the <110>center channel, and smaller than the

quadratic case for the <001>center channel (1/2): MAE

of 3.6 eV with the TTM ﬁtted compared to 13.4 eV in

the quadratic case. However, the TTM ﬁtted performs

clearly worse than the constant and quadratic EPH cases

for the <111>center channel, <001>oﬀ-center channel

(1/4) and <110>oﬀ-center channel trajectories.

Looking at Fig. 4(e), it is obvious that the worst agree-

ment between TTM ﬁtted and TDDFT is obtained for

the <001>oﬀ-center channel (1/4). Since the electronic

density reaches very high values in this case, the TTM

ﬁtted largely underestimates the value of the stopping

since it cannot vary with the electronic density contrary

to the stopping in the EPH model.

Now focusing on the incommensurate directions

of Fig. 5, the TTM curves in their SRIM-derived

parametrizations show again very large discrepancies

with the reference TDDFT data.

We also observe with the incommensurate directions

that during close encounters (collisions) between the pro-

jectile and an atom of the lattice, the energy lost to elec-

trons in the EPH model in the plots of Fig. 5is step-like.

This degrades the similarities between the TDDFT and

the EPH data. Due to this step-like behavior, rather than

searching for point-by-point correspondence between the

TDDFT data and our EPH curves during those close en-

counters, we aim to obtain a good ﬁt between EPH and

TDDFT data on average during the collision. In other

words, we aim to have the value of energy lost to electrons

with the EPH model as close as possible to the TDDFT

data, after the collision.

Despite this step-like behavior, visually from Fig. 5,

the comparison of diﬀerent incommensurate trajectories

seems to be in favor of the EPH model compared to the

TTM ﬁtted. Indeed, for all the sampled directions, the

EPH model, especially in its constant α(ρ) formulation,

follows the reference TDDFT curves quite closely, except

for the Incommensurate 3 direction (Fig. 5(d)) where the

energy lost during the collision (the electronic density

peak) is overestimated. The low density regions and in

most cases the high density regions seem to be well re-

produced by the EPH model, and the density dependence

of the energy lost to electrons globally well followed. It

can be easily noticed visually on Fig. 5(b) and Fig. 5(c)

that the constant MD-EPH performs better than the

quadratic one. Table III actually indicates that for in-

commensurate directions, the constant EPH always per-

forms better than the quadratic one.

According to Table III, the TTM ﬁtted performs worse

than the EPH constant case for all incommensurate direc-

tions, except for the incommensurate 3 case. The MAE

diﬀerence is the greatest for the incommensurate 2 di-

rection, with the EPH constant case having a MAE of

6.8 eV and the TTM ﬁtted a MAE of 29.2 eV. As in

the channeling trajectories, the case in which the TTM

ﬁtted performs the worst is the case in which high elec-

tronic density values are sampled. The quadratic EPH

reproduces the TDDFT data better than the TTM ﬁt-

ted for the incommensurate 1 (MAE of 9.3 eV compared

to 10.0 eV), the incommensurate 1 + vacancy (MAE of

15.8 eV compared to 18.4 eV), the incommensurate 2

(MAE of 14.5 eV compared to 29.2 eV) directions, but

worse for the incommensurate 3 (MAE of 26.1 eV com-

pared to 14.3 eV).

On the whole, the EPH constant and quadratic cases

11

TABLE III. Distance (MAE) of the EPH and TTM ﬁtted energy losses curves with respect to the reference TDDFT data.

EPH Constant (eV) EPH Quadratic (eV) TTM ﬁtted (eV)

Center channel <001>9.4 12.9 18.8

Center channel <110>57.9 67.2 35.7

Center channel <111>5.4 3.4 18.7

Oﬀ-center channel <001>(1/2) 4.2 13.4 3.6

Oﬀ-center channel <001>(1/4) 15.5 8.0 32.0

Oﬀ-center channel <110>3.5 2.5 19.8

Incommensurate 1 5.7 9.3 10.0

Incommensurate 1 + vacancy 8.2 15.8 18.4

Incommensurate 2 6.8 14.5 29.2

Incommensurate 3 21.5 26.1 14.3

Total 13.2 18.2 23.4

perform better than the TTM ﬁtted, with the total MAE

of the constant EPH case being of 13.2 eV, the one of the

quadratic EPH of 18.2 eV and the one of the TTM ﬁt-

ted of 23.4 eV. Visually, according to Fig. 4and Fig. 5,

the EPH model also allows to reproduce in many cases

very closely the density dependence of the stopping ob-

served in TDDFT simulations, which is, in addition to

the quantitatively better MAE, more physically satisfy-

ing than the always constant stopping given by the TTM

ﬁtted.

Also, on the whole, the constant EPH performs better

than the quadratic one (total MAE of 13.2 eV compared

to 18.2 eV). This can be surprising considering the rel-

ative complexity of the two functions, and the fact that

an α(ρ) parameter shape similar in many ways to our

quadratic expression was employed in [15]. This means

that the density terms appearing as weighting factors in

(3) are suﬃcient to guarantee a rich density dependent

behavior of the stopping in the EPH model. An impor-

tant reason for the less satisfactory performance of the

quadratic EPH with respect to TDDFT data lies in the

decay to zero we chose for the quadratic expression. In-

deed, in light of the graphs of Fig. 4and Fig. 5, this is

questionable, as no step-like behavior of the stopping is

observed in our TDDFT data as was the case in [15].

However, a decay is necessary otherwise the energy lost

to electrons would diverge in regions of high electronic

densities. As a compromise, it could be wise to set this

decay to a non-zero value instead of strictly zero.

Regarding the constant parametrization of the EPH

model, despite the good results it provides, the fact that

αis non-zero when ρ= 0 ought to be discussed. Indeed,

as we said earlier, it is physically more satisfactory to

have α(ρ)≈0 at low ρ. A consequence of this might be

an overestimation of the strength of the electron-phonon

coupling since the part of the α(ρ) function that gov-

erns the magnitude of electron-phonon coupling is the

low ρone. This is an aspect we can only evaluate by per-

forming MD-EPH simulations of collision cascades with

diﬀerent thermal parameters of the EPH model, as we

have done with the TTM in [13]. To provide an answer

to this question, among other objectives, the next section

is dedicated to simulations of collision cascades with the

EPH model and with the TTM for comparison.

V. MD SIMULATIONS OF COLLISION

CASCADES

A. TTM and EPH parameters employed in the

simulations

In this section, simulations of collision cascades in Si

with the TTM and the EPH model are carried out. With

the EPH model, cascades with the constant and the

quadratic parametrization of the parameter α(ρ) are per-

formed. With the TTM, we only ran cascades using the

SRIM derived parametrization of the TTM, as it is the

default and most widely employed way to parametrize the

TTM. Moreover, it is highly unlikely that TDDFT simu-

lations will be used to parametrize the TTM for collision

cascade simulations, as we did in the previous section.

The comparison between the results of the cascades with

the EPH model and with the TTM we perform in this sec-

tion should thus be envisioned more as a comparison be-

tween realistic usages of the models, rather than a purely

theoretical comparison between both models. If the elec-

tronic stopping parameter γsof the TTM is obtained

with SRIM, the electron-phonon coupling parameter γp

of the TTM is chosen on the basis of a careful literature

search, which led us to choose the value of the electron-

phonon coupling presented in [39]. The electronic stop-

ping parameter γsobtained with SRIM in Si is equal

to 39 g/mol/ps (LAMMPS metal units), which yields at

10 keV an electronic stopping power of about 11 eV/˚

A.

The value of the electron-phonon coupling γpthat we

use is 25 g/mol/ps (LAMMPS metal units). Considering

that in the TTM the total friction coeﬃcient acting on

atoms is the sum of γsand γp, both actually contribute

to the stopping [8]. At 10 keV, the γpparameter we use

yields an electronic stopping of about 7 eV/˚

A. In the fol-

lowing when we mention the eﬀects of electron-phonon

coupling in the TTM we do not refer to the eﬀects of

the γpparameter as part of the total friction force (elec-

12

tronic stopping eﬀects), but to the energy fed back from

the electrons to the ions via electron-phonon coupling,

whose magnitude is proportional to γp. We recall that

for the EPH model, no value of the electron-phonon cou-

pling must be speciﬁed because both electronic stopping

power and electron-phonon coupling are obtained from

the α(ρ) parameter.

To evaluate the inﬂuence of the thermal parameters of

the EPH model and of the TTM, i.e. the electronic spe-

ciﬁc heat Ceand the electronic thermal conductivity κe,

simulations with diﬀerent values of these parameters are

carried out. In all the simulations launched, Ceand κe

are considered constant. We employ this approximation

so that the eﬀects of Ceand κeare easier to observe and

discuss. In reality, Ceand κedepend on the electronic

temperature Te. Reliable Te-dependent values at low Te

for κecan be found for Si in [40], and can be extended to

Cewith the relation κe(Te) = ρDeCe(Te). For practical

purposes, for the calculation of Ceand κe, the electronic

diﬀusivity Deand the electronic density ρcan be con-

sidered constant and respectively equal to 20000 ˚

A2/ps

and 0.05 e−/˚

A3[41]. The procedure to obtain an expres-

sion for Ce(and κeconsequently) for each possible Teis

explained in [13].

Three scenarios are deﬁned:

•Scenario 1 refers to a case where Ceand κeare

deliberately set to low values compared to what

could be considered as the reference electronic tem-

perature Tedependent ones. In scenario 1, Ce=

5×10−6eV/K and κe= 5 ×10−3eV/K/˚

A/ps. It

corresponds to realistic values of Ceand κeat low

Te, but not on the entire range of Tespanned in

the simulations [13].

•Scenario 2 refers to a case where Ceand κeare pur-

posely set to high values compared to what could be

considered as the reference electronic temperature

Tedependent ones. In scenario 2, Ce= 3/2kB=

1.29×10−4eV/K and κe= 1.29×10−1eV/K/˚

A/ps.

It corresponds to realistic values of Ceand κeat

high Te, but not in the entire range of Tespanned

in the simulations [13].

•Scenario 3 refers to a mix between scenario 1 and

scenario 2: Cetakes the high value it has in scenario

1 and κethe low value it has in scenario 2. In

scenario 3, Ce= 1.29 ×10−4eV/K and κe= 5 ×

10−3eV/K/˚

A/ps.

These three scenarios are employed with the TTM

and with the constant parametrization of the EPH

model. Only scenario 3 is used with the EPH

quadratic parametrization, as our results with the con-

stant parametrization of the TTM show, as we will see

later in this section, that it is not necessary to perform

simulations with all the thermal parameters scenarios in

the quadratic EPH case.

Electronic grids made of 15 ×15 ×15 cubic cells are

employed both with the EPH model and the TTM.

B. Cascades properties and statistical quantities

The results are analyzed in terms of number of defects

(number of Frenkel pairs), number of clusters and PKA

penetration depth. The defects are counted with the

Lindemann sphere criterion [42] and a radius of 0.45 ˚

A,

which was demonstrated to yield for each cascade an al-

most constant ratio to the number of defects obtained

with the Wigner-Seitz method [43]. The proportional-

ity factor between the two methods that we ﬁnd in Si is

of 8, the same value was found by Nordlund et al. in

[43]. It is then possible to write, for Si, Nlin ≈8×Nw−s

where Nlin is the number of defects obtained with the

Lindemann method and Nw−sis the number of defects

obtained with the Wigner-Seitz method. Two defects

are considered to belong to the same cluster if they are

separated by a distance smaller than twice the nearest

neighbor distance, i.e. 4.7 ˚

A in Si. The PKA penetra-

tion depth is deﬁned in [33].

To quantify the number of defects, the number of clus-

ters and the PKA penetration depth, we use the mean

values calculated over the 75 calculations carried out for

each case studied, as well as graphical representations in

the form of box plots. In the box plots are represented

the minimum and maximum values of each set, as well as

the ﬁrst quartile, median (or second quartile) and third

quartile values of each set. The uncertainties of the mean

values are quantiﬁed with the calculations of the Stan-

dard Error of the Mean (SEM) quantities.

C. Results and discussions

Table IV summarizes the parameters employed in the

various collision cascades scenarios we simulated, and

gives the mean values of the number of defects, the num-

ber of clusters and the mean PKA penetration depth for

each of these scenarios. Fig. 6and Fig. 7respectively

show the distributions of the number of defects and of

the PKA penetration depth as box plots for all the stud-

ied simulation scenarios.

1. Defects and clusters evolution

From Fig. 6, the similarity between all the EPH sce-

narios is immediately striking. The diﬀerent thermal pa-

rameters employed in the constant case as well as the

switch between quadratic or constant parametrizations

do not lead to any visible eﬀects on the number of de-

fects. The values of the ﬁrst, second and third quartiles

are comparable between each studied scenario. The min-

imum and maximum values show larger discrepancies but

this cannot be attributed to changes in the thermal pa-

rameters or in the αfunction employed. The high degree

of stochasticity of the cascades makes it very diﬃcult to

obtain converged minimum and maximum values for a

given set of simulations. Consequently, it is very hard to

13

TABLE IV. Summary of the parameters employed in the diﬀerent sets of cascades simulations carried out and mean values

of the number of defects, the number of clusters and of the PKA penetration depth for all the simulation sets. The values in

brackets next to the mean values are the SEM values. Each mean value is calculated from a set of 75 simulations initiated with

distinct PKA directions. This table only gives a summary of the parameters employed, more details can be found in the text.

Model employed EPH TTM (SRIM)

αfunction (for EPH) Quadratic Constant

Thermal scenario name 3 1 2 3 1 2 3

CeHigh Low High High Low High High

κeLow Low High Low Low High Low

Elec. stopp. + el-ph Fitted Fitted Fitted Fitted SRIM + [39] SRIM + [39] SRIM + [39]

Mean number of defects 767 (21) 730 (17) 763 (21) 719 (16) 1299 (47) 473 (10) 555 (17)

Mean number of clusters 46 (1) 47 (1) 46 (1) 46 (1) 21 (1) 38 (1) 33 (1)

Mean PKA depth (˚

A) 156 (7) 160 (7) 151 (7) 150 (8) 151 (6) 150 (7) 144 (7)

draw conclusions from the minimum and maximum val-

ues only. The similarities in terms of number of defects

between all the scenarios studied with the EPH model

can also be observed in the mean values displayed in Ta-

ble IV, diﬀering by up to 48 defects and having SEM val-

ues of about 20 each time. The mean number of clusters

displayed in Table IV is even more similar, with mean

values diﬀering at most by one cluster only.

FIG. 6. Box plots of the number of defects for all the studied

sets of parameters with the TTM and EPH model. From

lowest to highest, the horizontal lines of a box plot represent

the minimum value of the set, the ﬁrst quartile value, the

median (or second quartile value, in yellow), the third quartile

and the maximum value of the set.

The fact that the constant and quadratic parametriza-

tions of the EPH model give similar defects number is

not surprising considering both parameters were ﬁtted

to reproduce the same reference TDDFT data. More in-

terestingly and less expected, the fact that the diﬀerent

thermal scenarios used in the constant αcase (which, as

we said earlier, should overestimate the magnitude of the

electron-phonon coupling), show no conclusive variations

in the number of defects reveals that the electron-phonon

coupling has no eﬀect on the defects formation. Indeed,

the thermal parameters Ceand κeof the EPH model

(and of the TTM), do not act on the friction force acting

on the moving projectile. They only control the tem-

poral rate of the electronic energy lost (Ce) to the ions

and the diﬀusion of the electronic energy to nearby elec-

trons (κe). Thus, in Si at 10 keV, no defect is created

due to the energy fed back from the electrons to the ions

via the electron-phonon coupling. In the standard SRIM

parametrization of the TTM, with the electron-phonon

coupling parameter taken from [39], we found in [13] that

electron-phonon coupling had an eﬀect on the creation of

defects in Si and Ge. In the above mentioned article, the

simulations were carried out at the lower temperature of

100 K. In this paper, we run new collision cascade sim-

ulations with the TTM with a thermostat temperature

of 300 K, as it is done for the EPH model. This allows

us to directly compare the results of the EPH and TTM

collision cascades.

By looking at the box plots of the TTM scenarios in

Fig. 6, it can be observed that the distributions of the

number of defects between the studied thermal scenarios

show impressive discrepancies. The entire distribution

of scenario 1 is obviously shifted upwards compared to

scenarios 2 and 3. Quantitatively, the median values in

scenarios 2 and 3 are respectively 467 and 517, while it is

1308 in scenario 1. The ﬁrst quartile values in scenarios

2 and 3 are respectively 404 and 453, while it is 994 in

scenario 1. The third quartile values in scenarios 2 and

3 are respectively 527 and 626, while it is 1559 in sce-

nario 1. This trend is conﬁrmed by the mean values of

the number of defects of Table IV, which are similar for

scenarios 2 and 3 (473 and 555 respectively) and about

3 times greater for scenario 1 (1383). This is in line with

the parametric study of the TTM we performed in [13],

where we observed that more defects were created with

low values of Ceand κe(in the cited article, κe=ρDeCe,

with Deand ρconstant). However, in the previous para-

metric study we performed on the TTM, the discrepan-

cies between similar scenarios for the thermal parameters

were much smaller. We checked the calculation of our

number of defects with the Weigner-Seitz method of the

OVITO software [44], and found the same factor of about

3 between the mean values. The fact that the present cal-

14

culations are carried out at 300 K instead of 100 K like

we did in [13] could explain these enhanced discrepancies.

As clearly explained in [13] and above in this article, the

only way the thermal parameters Ceand κecan inﬂuence

the number of defects is via the electron-phonon coupling.

If the energy fed back from the electrons to the ions via

electron-phonon coupling is enough to induce the melting

of the material, the number of observed defects increases.

Carrying out simulations at 300 K may favor this melting

behavior.

The comparison of the number of clusters also gives

valuable information regarding the melting of the mate-

rial in scenario 1 with the TTM. Indeed, according to

Table IV, we ﬁnd on average 38 and 33 clusters in the

TTM scenarios 2 and 3, and only 21 for scenario 1. Thus,

with scenario 1, more defects are contained in a smaller

number of clusters, which means that the clusters are

bigger. Looking at the size and shape of the clusters of

defects, in the case of scenario 1, for almost all the cas-

cades, the defects are almost all contained in very large

amorphous pockets, deﬁned as clusters containing more

than 100 defects. Those amorphous clusters are by far

less important in cascades of scenarios 2 and 3: 76% of

defects are contained in amorphous clusters in scenario

1 whereas only 10% and 25% in scenario 2 and 3 re-

spectively. The low values of Ceand κeindeed induced

melting via electron-phonon coupling, leading to these

large amorphous pockets.

From the basic heat diﬀusion equation of the TTM

[8], a low value of Ceresults in a high rate of energy ex-

change between electrons and ions [45], and a low value of

κeresults in a poor distribution of electronic heat in the

material. The electronic energy is contained in very spe-

ciﬁc regions of the material and is very quickly fed back

to the ions of the same regions, thus inducing melting.

From the ﬁgures mentioned above, it also appears that

a bit more defects are observed in scenario 3 compared

to scenario 2, as well as less clusters. It also indicates an

increase in melting of the material due to the low value

of κeemployed in scenario 3. However, this melting is

by far less important than in scenario 1. It means that

the combined eﬀects of Ceand κeare responsible for

the large melting observed in scenario 1. Note that with

the SW potential, the lattice thermal parameters of Si

(melting point and lattice thermal conductivity) are well

reproduced [46], the melting behavior of the material is

therefore realistic.

The important point to remember from those MD-

TTM cascades calculations in Si is that with the TTM,

electron-phonon coupling can have very large eﬀects on

the number of defects in collision cascades in Si. On the

contrary, with the EPH model, the electron-phonon cou-

pling was not found to have any eﬀect. Despite the fact

the parameters chosen for κeand Cein scenario 1 are

unrealistic, this is a ﬁrst clue pointing to the fact that

the TTM in its original form [8], and parametrized with

SRIM for γsand a literature search for γp, might over-

estimate the eﬀects of the electron-phonon coupling in

collision cascades. Another argument that supports this

point is that experimentally, amorphization of Si by elec-

tronic mechanisms occurs with swift irradiation of heavy

ions of considerably greater energy than our 10-keV col-

lision cascades, i.e. tens of MeV [47,48]. Thus, melt-

ing (amorphization) of the material via electron-phonon

coupling as important as the one we observe with the

TTM in scenario 1 is highly unrealistic. To counterbal-

ance what we just said, one could argue that the over-

all lower electronic stopping power at stake in the EPH

model compared to the TTM makes the amount of en-

ergy lost to electrons smaller than with the TTM and

thus reduces the possibility to observe electron-phonon

coupling eﬀects because there is less energy to give back

to the ions. MD-EPH simulations with various values

of constant αand performed on a system where thermal

eﬀects are more obvious than in Si, like Ge for example

[13,46], could help to shed light on this speciﬁc point.

We now focus on scenarios 2 and 3 of the TTM and 1,

2 and 3 for the EPH model, for which the energy trans-

fer via electron-phonon coupling has very few eﬀects on

the number of defects. The visual comparison of the

distributions of the number of defects shown in Fig. 6

with the EPH model and the TTM immediately reveals

that many more defects are created with the EPH model

than with the SRIM parametrization of the TTM. The

median of the number of defects for all scenarios in the

EPH model ranges from 719 and 763, whereas the TTM

scenario 2 and 3 have a median of 467 and 517. The

mean values of Table IV display roughly the same dis-

crepancy, with mean values ranging from 719 to 767 with

the EPH model and mean values of 473 (scenario 2) and

555 (scenario 3) with the TTM in its SRIM parametriza-

tion. This can be easily explained by the fact that the

amount of energy lost to the electrons obtained with the

EPH model ﬁtted to the reference TDDFT data is much

lower than the amount of energy lost to the electrons

with the SRIM parametrization of the TTM, as shown

in Fig. 4and Fig. 5. Consequently, as less energy is lost

to electrons, more energy is available to create defects via

collisions between ions or local melting of the matter as

it occurs in semiconducting materials [43,49]. This high-

lights the signiﬁcance of the existing controversy between

SRIM and TDDFT electronic stopping calculations, as

we prove here it has an important impact on the number

of defects created during collision cascades. To be com-

plete on this point, note that in the TTM, the electron-

phonon coupling parameter γpalso contributes to the

friction force, the total friction coeﬃcient being the sum

of the electronic stopping parameter γsand γp. The dis-

crepancies between the number of defects observed with

the EPH and the TTM scenarios are therefore enhanced

by this aspect of the TTM.

To conclude on the defects analysis, the results ob-

tained with the EPH model and with the TTM would

be much more similar if we had employed the TTM pa-

rameters we ﬁtted to the TDDFT data in Section IV.

However, a TTM parametrization with TDDFT data is

15

not representative of how the TTM is used in the liter-

ature. Here, we wanted to compare the results obtained

following the default guidelines for the parametrization

of both the EPH model [15] and the TTM [8].

2. PKA penetration depth

The analysis of the PKA penetration depth is more

straightforward than the number of defects and clusters.

The mean values of Table IV reveal that the mean PKA

penetration depth for the EPH scenarios are all com-

prised between 150 ˚

A and 160 ˚

A. The SEM values (about

8) accompanying these mean values lead to consider the

PKA penetration depth in all EPH scenarios as similar:

no evolution or trends in the mean PKA depth can be

observed. The fact that the thermal scenarios of the con-

stant parametrization of the EPH model do not give dif-

ferent values of the PKA penetration depth is expected,

as both Ceand κedo not act on the friction force to

which the moving ions are subjected. Considering the

similarities in the electronic stopping between the con-

stant and quadratic parametrization of the EPH model

for all directions studied in Section IV, the very similar

mean values of the PKA penetration depth for the EPH

constant and quadratic cases are also consistent.

The mean values of the PKA penetration depth for the

TTM scenarios are also very consistent (151 ˚

A, 150 ˚

A and

144 ˚

A for scenarios 1, 2 and 3 respectively), for the same

reasons as just exposed for the EPH thermal scenarios.

However, it seems that the mean values of the PKA pene-

tration depth of the EPH model are slightly greater than

the ones obtained with the SRIM parametrization of the

TTM. The uncertainties (SEM values) coming with those

mean values do not allow to be completely conclusive on

this point. The analysis of the box plots of Fig. 7do

not give much more information on this aspect as all the

distributions seem to be quite similar: no clear trend or

behavior can be observed.

An increase in the PKA penetration depth with the

EPH model would be physically sound. Indeed, the elec-

tronic stopping with the EPH model is overall smaller

than with the SRIM parametrization of the TTM we

adopted. Moreover, with the EPH model, the electronic

stopping is speciﬁcally low for channelling directions (see

Fig. 4), allowing for an increased depth of the PKAs if

they are themselves in a channelling direction. The slight

increase in the mean PKA penetration depth for the EPH

model is consistent with this, even if it is less blatant than

one might expect. If it is hard to be conclusive about the

impact of the electronic stopping on the PKA depth with

our simulations, it is obvious that the electronic stopping

can have an important impact on the creation of defects,

especially in semiconducting materials where the forma-

tion of amorphous pockets of defects can be observed.

FIG. 7. Box plots of the PKA penetration depth for all the

studied sets of parameters with the TTM and EPH model.

From lowest to highest, the horizontal lines of a box plot rep-

resent the minimum value of the set, the ﬁrst quartile value,

the median (or second quartile value, in yellow), the third

quartile and the maximum value of the set.

VI. CONCLUSION

First principles TDDFT calculations of electronic stop-

ping were combined with the EPH model for the inclu-

sion of non-adiabatic electronic eﬀects into MD simula-

tions of collision cascades. A simple parametrization of

the EPH model with a constant αfunction already gives

very satisfactory results in terms of electronic density

(or crystal direction) dependence of the electronic stop-

ping obtained with MD-EPH simulations in Si, consid-

ering TDDFT results as the reference. The TTM in its

SRIM parametrization obviously does not allow repro-

duction of the TDDFT data. However, when parameters

of the TTM are ﬁtted to the TDDFT data, although the

energy lost to electrons is independent of the electronic

density and the friction parameter is scalar, the results

obtained do not diﬀer as much as one could expect from

the TDDFT data. Nonetheless, the EPH model allows to

better reproduce the electronic stopping calculated with

TDDFT in Si than the TTM, whose incorporation of elec-

tronic stopping is not tensorial and density-dependent

as in the EPH model. Indeed, quantitatively the agree-

ment between the EPH model and TDDFT is better, and

qualitatively it is more satisfactory physically for the elec-

tronic stopping to be dependent on the electronic density.

By reﬁning the α(ρ) expressions employed for the ﬁtting,

we believe even better reproduction of the TDDFT data

can be achieved with the EPH model.

Collision cascades carried out in Si at 10 keV have

revealed that the choice of the model employed for the

inclusion of the electronic eﬀects and the parametrization

16

of the chosen model have a signiﬁcant inﬂuence on the

number of defects created. Our collision cascade simula-

tions performed with the EPH model ﬁtted on TDDFT

data give signiﬁcantly more defects than cascades carried

out with the TTM parametrized with SRIM calculations.

Fundamental discussions comparing SRIM and TDDFT

electronic stopping calculations are then of prime impor-

tance as it has visible eﬀects on collision cascades even

at the relatively low energy of 10 keV. The impact of

the electron-phonon coupling on the creation of defects

is found to be signiﬁcant with the TTM and highly de-

pendent on the Ceand κeparameters chosen, whereas

no inﬂuence of the electron-phonon coupling on the cre-

ation of defects was observed with the EPH model in Si.

This last aspect indicates a likely overestimation of the

signiﬁcance of electron-phonon coupling in the naive but

widespread parametrization method we employed for the

TTM. Additional calculations with the EPH model at

diﬀerent energies, with diﬀerent parameters and on dif-

ferent systems would considerably help to be fully con-

clusive on the signiﬁcance of electron-phonon coupling in

collision cascades.

We believe this paper gives strong evidence in favor of

the necessity for further studies comparing SRIM-derived

and TDDFT-derived electronic stopping (actually on the

understanding and quantiﬁcation of electronic stopping

in general). In particular, the issue of the charge state

of the projectile, which was not addressed in this article,

should be addressed in the future as it may have sig-

niﬁcant quantitative implications on the stoppings cal-

culated with TDDFT. In addition, this paper provides

trustworthy arguments defending the signiﬁcant impact

the ﬁner incorporation of electron-phonon coupling in the

EPH model has on the results of collision cascades.

ACKNOWLEDGMENTS

The authors would like to thank Andre Schleife

and Cheng-Wei Lee from University of Illinois Urbana-

Champaign as well as Alfredo Correa and Artur Tamm

from Lawrence Livermore National Laboratory for fruit-

ful discussions. The authors would also like to thank

Chlo´e Simha for proofreading this manuscript. This ar-

ticle is based upon work from COST Action TUMIEE

CA17126, supported by COST (European Cooperation

in Science and Technology). Calculations have been

performed using HPC resources from GENCI-CCRT

supercomputer at CEA, DAM, DIF, HPC resources

from GENCI (Grant A0030907474) and HPC resources

from CALMIP (Grant 1555). J. Teunissen and F. Da

Pieve have received funding from the Research Executive

Agency under the EU’s Horizon2020 Research and Inno-

vation program, project ESC2RAD (grant ID 776410).

T. Jarrin, A. H´emeryck and N. Richard are active mem-

bers of the Multiscale and Multi-Model Approach for Ma-

terials in Applied Science consortium (MAMMASMIAS

consortium), and acknowledge the eﬀorts of the consor-

tium in fostering scientiﬁc collaboration.

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