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The effects of electron transfer on the energy loss of slow He 2+, C 2+, and C 4+

ions penetrating a graphene fragment

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2014 J. Phys.: Condens. Matter 26 085402

(http://iopscience.iop.org/0953-8984/26/8/085402)

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Journal of Physics: Condensed Matter

J. Phys.: Condens. Matter 26 (2014) 085402 (7pp) doi:10.1088/0953-8984/26/8/085402

The effects of electron transfer on theenergy loss of slow He2+, C2+, and C4+

ions penetrating a graphene fragmentFei Mao1,2, Chao Zhang1,2, Cong-Zhang Gao1,2, Jinxia Dai1,2 andFeng-Shou Zhang1,2,3

1 The Key Laboratory of Beam Technology and Material Modification of Ministry of Education,College of Nuclear Science and Technology, Beijing Normal University, Beijing 100875,People’s Republic of China2 Beijing Radiation Center, Beijing 100875, People’s Republic of China3 Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator of Lanzhou,Lanzhou 730000, People’s Republic of China

E-mail: fszhang@bnu.edu.cn

Received 30 August 2013, revised 17 December 2013Accepted for publication 2 January 2014Published 7 February 2014

AbstractElectronic energy loss in the collision processes of slow ions with a graphene fragment isinvestigated by combining ab initio time-dependent density functional theory calculations forelectrons with molecular dynamics simulations for ions in real time and real space. We studythe electronic energy loss of slow He2+, C2+, and C4+ ions penetrating the graphene fragmentas a function of the ion velocity, and establish the velocity-proportional energy loss forlow-charged ions down to 0.1 a.u. One mechanism clarified in the simulations for electrontransfer is polarization capture, which is effective for bare ions at low velocities. The other oneis resonance capture, by which the incident ion can capture electrons from the graphenefragment to its electron affinity levels, which have the same, or nearly the same, energy asthose of the electron donor levels. The results demonstrate that the nonlinear behavior ofenergy loss of C4+ is attributed to the large number of electrons captured by this multi-chargedion during the collision.

Keywords: graphene fragment, electronic stopping, TDDFT-MD, electron transfer

(Some figures may appear in colour only in the online journal)

1. Introduction

Low-velocity ion interactions with solids are an active field offundamental science, and the stopping power (SP) is importantfor many technological applications, such as in shallow ion im-plantation, radiation physics, and radiotherapy [1–12]. Whenan ion propagates through matter, it gradually deposits energy,either by inelastic processes leading to electronic excitations,or elastic collisions with lattice atoms of the host materials.Both experimental and theoretical studies of the SP for slowions (v ≤ 1.0 a.u.) show that the main stopping mechanism inmetals is due to the excitation of conduction electrons near theFermi energy level [1–4]. In the low-energy regime, theories

of SP in metals are built on the assumption that the electronscan be approximated by a homogeneous electron gas. Thefree-electron gas model is applied, and yields the followingexpression for the velocity dependence of the SP,

dE/dx =−Qv, (1)

with a friction coefficient Q, where v is the ion velocity.However, measured SP for slow protons in various noblemetals (Au, Ag and Cu) showed unexpected deviations fromthe proportionality to ion velocity in the range of low energies[1]. These deviations have been interpreted as a thresholdeffect in the excitation of d electrons, which usually begin

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J. Phys.: Condens. Matter 26 (2014) 085402 F Mao et al

to contribute to the SP in transition metals at intermediateenergies (about 0.1≤ v ≤ 0.3 a.u.) [1–4].

The question of threshold effects in low-energy stoppinghas drawn great interest recently. For solid insulators, thelow-energy electron excitations are expected to be suppresseddue to a minimum excitation energy (energy band gap) of thematerials, implying that a threshold effect in the electronicSP could be expected [5, 6]. However, earlier experimentalresults showed that when protons traversed a thin film ofthe wide-band-gap insulator LiF, no threshold value on thevelocity dependence of electronic stopping cross section wasfound [13]. Meanwhile, a threshold-like behavior was reportedfor the energy loss in grazing scattering of hydrogen ions froma LiF surface [14], and charge exchange between the ion and Fanion of the crystal along the trajectories has been identified asthe dominant energy loss mechanism in this case. Recently, thelinear velocity dependence of the electronic SP was observedfor protons backscattered from LiF, and an apparent velocitythreshold at 0.1 a.u. was verified [15–17]. A velocity thresholdat about 0.1 a.u. was also measured for protons transmittedthrough AlF3 thin films [17, 18].

An interest in understanding the complexity of thishighly nonequilibrium process has attracted a huge amount ofresearch, both experimentally and using computer simula-tions, aimed at understanding radiation damage in matter.For projectiles faster than the Fermi velocity of the targetelectrons, perturbation models apply, such as those based onBethe–Bloch theory [19]. At low energies, where perturbationtheory fails, a fully self-consistent treatment of the electronicstructure of the combined projectile–target is required. Thestudy of SP differences is far from understood. In order toaccurately calculate the ion energy loss by inelastic collisionwith bound electrons of the target, to the best of our knowledge,ab initio principles are the best method to simulate thisnon-adiabatic process, in which the electronic structure ofthe stopping materials has been taken into account. Manyworks have tested the validity of this method for simulatingthe electronic excitation and calculating the energy loss [4, 5,11, 12, 20–22]. In order to establish the validity limits for theBorn–Oppenheimer approximation and the threshold energyfor single-vacancy defect formation in carbon nanostructures,Krasheninnikov et al [12] simulated the collision betweenenergetic protons and graphitic carbon nanostructures. Toexplore the defect formation mechanisms, Bubin et al [21]calculated the amount of energy obtained by the graphenefragments after proton collision.

Actually, there is also a minimum excitation energy in themolecular target system; the energy gap between the highestoccupied molecular orbital (HOMO) and lowest unoccupiedmolecular orbital (LUMO). In this respect, the electronicstopping for the molecular target is expected to behave likethose of solid insulators. The purpose of this paper is to clarifywhether there is a departure from the proportional dependenceof the SP on the ion velocity. One question of basic interestwhich has not been adequately studied so far is the mechanismof electron transfer happening in the ion–molecule collision[23, 24] or in the ion–surface collision [22, 25–27]. Themechanisms of how the ion multi-charged state affect theelectronic energy loss are investigated in this work.

The structure of the paper is as follows. In section 2 webriefly introduce the theoretical framework and the computa-tional details of this work. In section 3 a number of calculatedresults are shown and discussed. Conclusions are presented insection 4.

2. Method and computational details

In the present work, in order to characterize the collisionbehaviors of He/C ions with the nuclei and electrons of thegraphene fragment [28], the Ehrenfest molecular dynamicstime-dependent density functional theory (MD-TDDFT) (E-TDDFT) model [4, 7, 29–32] is employed to study thesecollisions. In this theoretical framework, the electron dynamicsis described by ab initio time-dependent density functionaltheory and the nuclei are approximated as classic ions. Thismethod allows for excited electronic states ab initio MD(AIMD) simulation. The Ehrenfest MD scheme [33, 34] isdefined by the following coupled differential equations (atomicunits are used here):

i∂9(x, t)∂t

= He(r, ER(t))9(x, t), (2)

MJd2 ERJ

dt2 =−

∫dx 9∗(x, t)[∇J He(r, ER(t))]9(x, t)

−∇J∑I 6=J

Z I Z J

| ERI (t)− ERJ (t)|(3)

where 9(x, t) is the many-body electron wavefunction, wedefine x ≡ {x j }

Nj=1 and x j ≡ (Er j , σ j ), which include the

coordinates Er j and the spin σ j of the j th electron, N isthe number of electrons in the system. He(r, ER(t)) is theelectronic Hamiltonian and ER(t)≡ { ER1(t), . . . , ERM (t)} is theinstantaneous distribution of all the nuclear positions, and Mis the number of nuclei in the system. The motion of the nucleiis determined by the set of equations in (3). The electronicHamiltonian is given by:

He(r, ER(t))=−N∑i

12∇

2i +

∑i< j

1|Eri − Er j |

−

∑i J

Z J

| ERJ − Eri |.

(4)

This form of the Hamiltonian allows one to write the forcethat acts on each nucleus in terms of the electronic densityn(Er , t), so we can rewrite equation (3) as:

MJd2 ERJ

dt2 =−

∫dEr n(Er , t)[∇J He(r, ER(t))]

−∇J∑I 6=J

Z I Z J

| ERI (t)− ERJ (t)|. (5)

Equation (5) shows that the ionic force can be calculatedfrom the time-dependent electronic density n(Er , t). This fact isthe basis for TDDFT-based Ehrenfest MD. Instead of solvingequation (2), we can solve the corresponding time-dependent

2

J. Phys.: Condens. Matter 26 (2014) 085402 F Mao et al

Kohn–Sham (KS) system, which provides an approximationto n(Er , t):

i∂ϕi (Er , t)∂t

=

[−

12∇

2+ υKS[n](Er , t)

]ϕi (Er , t),

i = 1, . . . , N , (6)

n(Er , t)= 2N/2∑i=1

|ϕi (Er , t)|2 (7)

where ϕi (Er , t) is the KS electron orbital, υKS(Er , t) is thetime-dependent KS potential,

υKS(Er , t)= υext(Er , t)+ υH(Er , t)+ υxc(Er , t) (8)

with

υext(Er , t)=−∑

J

Z J

| ERJ (t)− Er |(9)

υH(Er , t)=∫

dEr ′n(Er ′, t)|Er − Er ′|

, (10)

where υH(Er , t) is the Hartree potential, and υext(Er , t) andυxc(Er , t) are the time-dependent electron–nucleus andexchange–correlation potential, respectively. In this model,the potential energy and forces acting among the ions arecalculated ‘on the fly’ as the simulation proceeds. In Ehren-fest MD, transitions between electronic adiabatic states areincluded, and it couples the populations of the adiabatic statesto the nuclei trajectories [34]. It opens a way to study electrontransfer between the ion and the target electrons during thecollision [23, 35].

In the present calculation, the interactions between theionic cores and valence electrons are represented by a norm-conserving Troullier–Martins (TM) pseudopotential, andυxc(Er , t) is approximated as the simplest form, the adiabaticlocal density approximation. From the computational pointof view, an excellent advantage of combining TDDFT withEhrenfest MD is that it provides an orthogonalization-freeAIMD calculation, which saves a lot of computer resourcesand time. In this paper, the calculations are performed by usingthe OCTOPUS ab initio code [36, 37]. The approximatedenforced time-reversal symmetry method [38] is employedto propagate the electronic wave functions, and the Verletalgorithm is used for the ionic motion equations.

The graphene fragment is composed by 72 atoms (includ-ing 54 carbon atoms, with 18 hydrogen atoms passivated atthe edge of the fragment). The simulation cell is a rectangularbox, the dimensions of which are 30 Å× 30 Å× 80 Å. Thetarget is placed in the xy plane, the distances between the xand y boundaries and the fragment are larger than 5 Å. Inour simulations, the collision trajectories for He2+ and C4+

start at 60 Å above the target and end at the same distancepast the sheet. For C2+, as the ion initially carries two valenceelectrons, its start position is located at the boundary of thesimulation cell (38 Å apart from the target plane). The externalpotential, electron density and KS orbitals are discretized in aset of mesh grid points with a uniform spacing of 0.18 Å alongall three spatial coordinates in real space.

Figure 1. Energy deposited in the electronic subsystem of the targetas a function of velocity for He2+, C2+, and C4+ ions, respectively.The dotted lines are given to guide the eye to the intercepts of thelinear extrapolation. See more details in the text.

In order to study the dynamics of the electronic subsystem,we have to choose a region with a larger electron density. Wechoose the middle of a sp2 bond as the impact point in thegraphene sheet. Before the time-dependent calculations, thevalence electrons of the whole system (including the projectileand the target) should be in their own ground states. Theground state orbitals are calculated by diagonalization of thetime-independent KS Hamiltonian. In the time evolution, thenuclei of the target are fixed [4, 11]. There are two reasonsto account for this: (i) what we are concerned about is theenergy obtained by the electronic subsystem of the target, andthe nuclear stopping can be excluded from the energy loss inthis way; and (ii) it is not desirable that an atomic defect isintroduced in the graphene by the ion collision, otherwise thevariations of electronic energy are not exclusively caused bythe electronic stopping of the projectile.

3. Results and discussion

Figure 1 shows the electronic energy loss versus ion velocity.The results exhibit a velocity-proportional energy loss forprojectile He2+ and C2+, which is consistent with the well-established relation of the electronic SP −dE/dx ∼ v inmetals. The energy gained by the electronic subsystem reachesa maximum value of 160 eV for He2+ at 0.9 a.u., which agreeswell with the results reported in the literature [21]. In the lightof the velocity-proportionality assumption, we extrapolate theenergy loss data down to zero incident energy (see the dottedlines in figure 1), and then the corresponding energy lossesfor He2+ and C2+ are 25 and 137.4 eV, respectively. This isobviously not reasonable, because there should be no energyloss when the ions do not collide with the graphene fragment atall. The extrapolation results imply that there exists a thresholdvelocity for this velocity-proportional SP. We can expect thatthe SP falls faster than linearity to zero below this thresholdvelocity. Besides, the results for C4+ deviate a little from thelinear dependence in the whole energy range.

With respect to target electron dynamics, the collisionprocess is a non-adiabatic, nonequilibrium and local process.

3

J. Phys.: Condens. Matter 26 (2014) 085402 F Mao et al

Figure 2. The two-dimensional (2D) electronic charge density evolution of the graphene sheet during the collision with He2+ at 0.6 a.u. (a)t = 0 fs, the electrons of the target are in their ground states. (b) t = 4.530 fs, the ion is penetrating the bond middle of the graphene sheet.(c) t = 4.681 fs, the ion passes the sheet. (d) t = 4.756 fs, the electron density almost reaches equilibrium just after the collision. Thesimulation time is appended in the figures.

By inspecting the time evolution of the electron density of agraphene sheet (see figure 2), we find that the energy transferis a local process. After the collision, the electrons becomeexcited and the electron density is disturbed. In this way,non-adiabatic energy transfer into the electronic subsystemis achieved, and it is spread by transport processes until theelectronic system reaches equilibrium again, and finally itbecomes thermal energy. The electron–electron interaction isresponsible for the recovery of thermal equilibrium of theelectronic subsystem. In the time evolution, it is shown that thedistribution of electron density reaches equilibrium again veryrapidly after the collision, meaning that the electron responseis very quick. More detailed information about the electronicexcitations in the target can be obtained by analyzing theoccupations of the KS states of the target system.

The electron transfer processes for C4+ ion are displayedby the electron density evolutions in figure 3. By integratingthe valence charge density around the ion nucleus after thecollision, we can get the number of electrons captured bythe ion—the results of which are shown in figure 4. Fora given velocity, it is shown that the number of electronscaptured by the C4+ ion is greater than those of He2+ andC2+ ions. Compared to He2+ and C2+ ions, the number oftransferred electrons is much more strongly related to the ionvelocity for C4+. We have checked the velocity dependenceof this charge-state-dependent electron transfer, and there aretwo mechanisms responsible for the electron transfer in oursimulations. In the simulations of He2+ and C4+ bare ioncollisions, the ions can polarize the electron cloud of thetarget molecule and then capture electrons from it when the

ions approach the target plane close enough (see figure 3).However, this mechanism of electron transfer is ineffective inthe low-velocity range, because only in the low-energy casehas the ion enough time to polarize the electrons away from thetarget plane and capture them. We call it polarization capture(PC), in which the number of transferred electrons is dependenton the velocity. One can see that the number of transferredelectrons decreases rapidly for the C4+ ion when the velocityis greater than 0.4 a.u. The PC process is not observed forC2+ in our simulations. Because of the screening effect of thevalence electrons, C2+ cannot polarize the electron cloud ofthe target or capture electrons from it.

Another mechanism for electron transfer in the ion–molecule collisions is ascribed to resonance capture (RC), inwhich the projectile ion captures the electrons from the targetto its atomic levels that have the same or nearly the sameenergy eigenvalues as those of the electron donor levels. Thismechanism was employed to explain the formation of H− ionsin grazing scattering of hydrogen on a metal Al surface [39].According to Koopmans’ theorem, the ionization potentialis taken to be equal to the negative of the orbital energy ofthe least-bound electron in the atom. In our simulations, theeigenvalue calculated from the pseudopotential for the 1s stateof the helium atom is about −17.78 eV, so we consider 17.78eV as the ionization energy of He2+ in this text. The staticcalculation shows that the ground state eigenvalue of HOMO inthe graphene fragment is −5.0 eV. Actually, by inspecting theKS eigenvalues of the graphene fragment during the collisionprocess, we find that all the KS eigenvalues are disturbed bythe ion collision.

4

J. Phys.: Condens. Matter 26 (2014) 085402 F Mao et al

Figure 3. Time evolution of the three-dimensional electron density of a C4+ ion collision with a graphene fragment at 0.1 a.u. (a)t = 25.57 fs, the C4+ is a bare ion before the collision. (b) t = 25.83 fs, the ion begins to polarize the electron distribution of the target whenit approaches more closely. (c) t = 26.09 fs, the electron density is polarized away from the molecular plane. (d) t = 26.36 fs, the electronsare transferred to the ion. (e) t = 28.73 fs, the electrons attached to the ion are about to leave the donor. (f) t = 30.04 fs, the electron transferprocess ends. See more details in the text.

Figure 4. Number of electrons captured from the target to theprojectile as a function of velocity for He2+, C2+, and C4+ ions,respectively.

For He2+ at 1.0 a.u., the time evolution of the seven highestoccupied KS states of the graphene fragment is shown infigure 5. One can see that they are compressed to lower valueswhen the ion approaches the graphene fragment, and after the

Figure 5. Eigenvalues of the seven highest occupied states as afunction of time for He2+ at 1.0 a.u. See details in the text.

ion penetrates the layer (at about 2.74 fs in the simulationtime) they rise again due to the electron excitations, whichare consistent with the results of Ar7+ ions traveling througha graphene sheet [22]. However, the eigenvalues of severalhigh-lying KS states are always within the ionization energy

5

J. Phys.: Condens. Matter 26 (2014) 085402 F Mao et al

Figure 6. Electron occupation number distribution after the collisionends. The left side of the dotted line shows the occupation of theground state and the right shows the excited states. This is the caseof a He2+ ion at 1.0 a.u. See details in the text.

of the helium ion in the collision process. It is possible for theion to capture electrons from these KS orbitals by resonanceneutralization when it penetrates the target electron cloud. Notethat the electrons are not only captured by the 1s state of theHe2+ ion, but also transferred into higher atomic orbitals, sothe resonant energies corresponding to those capture processesare higher than −17.78 eV.

We perform occupation analysis for all the KS statesof the graphene fragment after the collision. The electrondistribution in the excited states is defined by the projectionof the time-dependent wavefunction onto the initial particlestates (including a number of unoccupied states) extended tothe excitation energy range [40, 41],

nocc(m, k)=∑

n′|〈ϕmk |ϕn′k(T )〉|2, (11)

where m represents the particle states, n is the band index andk is the Bloch wavevector. The occupation number distributionfor the He2+ ion at 1.0 a.u. is shown in figure 6. It isfound that in this case there are electron losses in most ofthe ground states, about 13.6 electrons are excited, in which0.76 electrons are verified to be captured by the ion, andthe remaining 12.84 electrons are considered to transit to theexcited molecular orbitals of the graphene fragment (includingthe electrons which are supposed emitted). We emphasizethat KS spectra are not true spectra, and transitions betweenoccupied and unoccupied KS orbitals can be regarded as a firstapproximation to the true excitations of the system [42–44];in particular, the LDA HOMO approximates the true HOMOvery well.

The deviations from the velocity-proportional SP are at-tributed to electron transfer in the collision processes, becauseelectron transfer, being an important energy loss channel, isnot included in the linear SP theory. The velocity-linear SP inequation (1) is based on the assumption that the SP in solidsresults from the electronic excitation only. In our simulations,the PC process is dependent on velocity. It can be expected

that the electron transfer channel becomes more important forenergy loss in the even lower energy range (v ≤ 0.1 a.u.),and it should be responsible for the nonlinear effects belowthe threshold velocity of the SP proportional dependence.However, compared to electron excitation, further studies arerequired to determine to what extent the electron transferprocess affects the energy loss.

4. Conclusions

In conclusion, by using time-dependent density functionaltheory combined with molecular dynamics simulations, wepresent a non-perturbative first-principles study of the elec-tronic energy loss for ions (He2+, C2+, and C4+) whenthey penetrate a graphene fragment. Our results reveal thatvelocity-proportionality for the electronic energy loss of He2+

and C2+ ions is valid down to 0.1 a.u., but this relationshipwill lose validity at much lower energies. It implies thereis a threshold effect in the velocity-proportional electronicstopping of a graphene fragment.

Our results show that electronic excitation and electrontransfer are the most important channels for energy loss,and are responsible for the energy gained by the electronicsubsystem of the graphene fragment. We have also clarifiedthe mechanisms for both processes. Electronic excitation istriggered by the ion–electron collision. For a low-charge ion,the electronic energy loss exhibits a linear dependence on theion velocity. However, for a multi-charged ion, it departs froma perfect linear dependence. This is related to the large numberof electrons captured by the ion in the collision process.

The electron transfer processes are explained by twomechanisms in our simulations. Firstly, at low velocities(≤0.4 a.u.), when the bare ion is near and above the moleculesheet, it can capture electrons from the graphene fragment bypolarization interaction. Secondly, when the affinity levels ofthe positively charged incident ions lie in the valence states ofthe graphene fragment, it can obtain electrons from the HOMOand sub-HOMOs of the molecule via RC or near-RC when itcollides with the electrons of the graphene fragment.

Acknowledgments

The authors thank F Wang and F Calvayrac for fruitfuldiscussions. This work was supported by the National NaturalScience Foundation of China under grant nos 11025524 and11161130520, National Basic Research Program of Chinaunder grant no. 2010CB832903, and the EuropeanCommission’s 7th Framework Programme (FP7-PEOPLE-2010-IRSES) under grant agreement project no. 269131.

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