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Integration of electronic effects into molecular dynamics simulations of collision cascades in silicon from first-principles calculations

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Abstract

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.
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
IX
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= 52/2×alat for the <110>direction, and
x=3/2×alat and x= 53/2×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 rst step
Calculate distance D between two
sets of runs
D < cuto ?
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 (+2)×(1 + ec(ρρf))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(ρ0.3) 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=
5×106eV/K and κe= 5 ×103eV/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=
1.29×104eV/K and κe= 1.29×101eV/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 ×104eV/K and κe= 5 ×
103eV/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×Nws
where Nlin is the number of defects obtained with the
Lindemann method and Nwsis 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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... Many examples of accurate first-principles calculations of the electronic stopping power are available in the literature [117,118,[163][164][165][166][167][168]. Recent studies have demonstrated that electronic excitations (induced by both the primary impacting ion and especially by PKAs and further displaced atoms) affect the cascade evolution [118,[169][170][171] and thus, they need to be accounted for. The electronic stopping effects can be included in MD cascade simulations through the so-called twotemperature (2T) model [118,172]. ...
... Recent studies have extended the 2T model by coupling the electronic and nuclear effects via many-body forces that act in a correlated way. This allowed for the construction of a unified model for ionelectron interactions [170,171,173,174] with a complex energy-exchange process between the ionic and electronic subsystems [174]. ...
... On a parallel research stream, multiscale studies in a number of materials combining MD simulations of collision cascades with the electronic stopping from TDDFT offer a more accurate description of both the number and the nature of defects created under realistic conditions. The electronic degrees of freedom and their coupling to the phonons of the target affect the cascade evolution and morphology [170,171,173,174]. This is of relevance for the NIEL which includes a part of energy dissipated to phonons. ...
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