PHYSICAL REVIEW A 89, 022707 (2014)

First-principles study of the threshold effect in the electronic stopping power of LiF and SiO2 forlow-velocity protons and helium ions

Fei Mao,1,2 Chao Zhang,1,2 Jinxia Dai,1,2 and Feng-Shou Zhang1,2,3,*

1The Key Laboratory of Beam Technology and Material Modification of Ministry of Education, College of Nuclear Science and Technology,Beijing Normal University, Beijing 100875, China2Beijing Radiation Center, Beijing 100875, China

3Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator of Lanzhou, Lanzhou 730000, China(Received 4 November 2013; published 19 February 2014)

Nonadiabatic dynamics simulations are performed to investigate the electronic stopping power of LiF andSiO2-cristobalite-high crystalline thin films when protons and helium ions are hyperchanneling in the 〈001〉 axis.In this theoretical framework, ab initio time-dependent density-functional theory calculations for electrons arecombined with molecular dynamics simulations for ions in real time and real space. The energy transfer processbetween the ions and the electronic subsystem of LiF and SiO2 nanostructures is studied. The velocity-proportionalstopping power of LiF and SiO2 for protons and helium ions is predicted in the low-energy range. The measuredvelocity thresholds of protons in LiF and SiO2, and helium ions in LiF are reproduced. The convergence ofthe threshold effect with respect to the separation of grid points is confirmed. The underlying physics of thethreshold effect is clarified by analyzing the conduction band electron distribution. In addition, the electrontransfer processes between the projectile ions and solid atoms in hyperchanneling condition are studied, and itseffects on the energy loss is investigated.

DOI: 10.1103/PhysRevA.89.022707 PACS number(s): 61.85.+p, 31.15.A−, 61.80.Az, 61.82.Ms

I. INTRODUCTION

Electronic interaction between slow light ions and solidsis important in many fields of fundamental research andtechnology applications, such as shallow ion implantation,plasma physics, and material science [1–3]. Recently, thequestion of the threshold effects in low-energy (v < v0, v0 isthe Bohr velocity) electronic stopping has raised great interest.Theories of electronic stopping power (SP) in the low-energyregime are built on the assumption that the electrons inthe solid can be approximated by a homogeneous electrongas (HEG). The main stopping mechanism in metals is theexcitation of the conduction electrons near the Fermi energylevel. In low-energy range, both theoretical predictions andexperimental measurements [4] have shown that the electronicSP (−dE/dx) of light ions is proportional to the ion velocity:

− dE/dx ∝ v. (1)

Unexpectedly, the results of the very recent experimentalstudies on the electronic energy loss of H and He ionstransmitted a HEG-like metal Al film showed that, the SPvalues for helium ion exhibit a different velocity dependencein this low-energy region. It is explained by an additionalcontribution from the charge exchange occurring between Heions and Al, which is also considered as an important energyloss channel in the ion-solid interactions [5].

The interactions of slow ions with electronic systems thathave a finite minimum excitation energy Tmin, such as noblegas, reveal a threshold effect in the electronic SP. The stoppingof slow light ions in nobel gas should clearly differ from thatin metals, where a finite energy is required for the excitationof the outermost electrons [6,7]. In the collision of protons

*Corresponding author: fszhang@bnu.edu.cn

with atomic hydrogen, helium, and neon target, a low-energythreshold in the electronic energy loss is found, and it isascribed to the quantization of the target energy levels [8].Recently, theoretical and experimental studies have shownthat similar effects exist in noble metals and large bandgap insulator targets, where substantial electronic excitationenergies are required to excite the inner-shell electrons or thevalence band electrons, giving rise to threshold effects forprojectile stopping in these materials [9].

A special attention is paid to the electronic stopping of lightions with low velocities in nobel metals (Au, Ag, and Cu). Innoble metals, the measured SP for slow protons has shownunexpected deviations from the velocity proportionality, and acomplex structure of the SP versus ion velocity is observed inthe low-energy range [10,11]. Since in these metals the densityof states of d electrons, formed by an intense and narrow band,is located by a few eV with respect to the Fermi energy; thenumber of electrons contributing to the energy loss increaseswhen the projectile velocity gets high enough to excite thed electrons, thus leading to a complex dependence of SP onthe ion velocity. Recently, the electronic SP for low-energyions in gold has been carefully investigated [12,13]; the resultsreproduce a steeper slope in the proportional energy loss whenion velocity v > 0.18 a.u. by an explicit consideration of theexcitation of d electrons in theoretical models.

For insulators, the low-energy electronic excitations areexpected to be suppressed due to the energy band gap (Eg) ofthe materials, which results in a threshold effect for projectilestopping with respect to the ion velocity [14,15]. However,earlier experimental measurements showed that when theprotons are traversing polycrystalline lithiumfluoride (LiF) andsilica (SiO2), no threshold value in the velocity dependence ofelectronic stopping cross section (SCS) was found [16,17]. Thedeviation from the linear dependence of the SP was observedin grazing scattering of protons from the flat LiF surface [18].

1050-2947/2014/89(2)/022707(8) 022707-1 ©2014 American Physical Society

FEI MAO, CHAO ZHANG, JINXIA DAI, AND FENG-SHOU ZHANG PHYSICAL REVIEW A 89, 022707 (2014)

The charge exchange between the projectile ion and F− ofthe crystal along the trajectories has been identified as thedominant energy loss mechanism. The velocity proportionalityof the electronic SP was observed when protons and heliumions backscattered from LiF, SiO2, and KCl thin films, andan apparent velocity threshold at 0.1 a.u. for protons wasconfirmed in these large band-gap insulators. However, whatis puzzling is that no threshold effect was detected for heliumions traveling in SiO2 [15]. More recently, a velocity thresholdat about 0.1 a.u. was measured for protons passing throughAlF3 thin films [19,20].

For the theoretical aspect, calculations based on molecularorbital correlation diagrams for the H/LiF collision systemare performed. The lack of threshold values in the velocitydependence of electronic SCS was explained as a localreduction of the band gap due to the presence of the projectileion [16]. Pruneda et al. [21] performed the calculations whichare based on an ab initio method, and the results revealed thata velocity threshold is at 0.2 a.u. for proton traveling along the〈110〉 channel in LiF, which is still higher than that observed inexperiments. In Ref. [22], Solleder et al. calculated the min-imum excitation energy of LiF perturbed by the (anti)protonimpurity by employing the multiconfiguration self-consistentfield method; the results showed that the excitation energy ofLiF was significantly reduced by the (anti)proton with respectto the unperturbed crystal. In Refs. [19,20], a model based ona restricted transport cross section and DFT was employed toinvestigate the SP behavior in LiF, and a velocity threshold at0.1 a.u. for proton is predicted by introducing a finite energygap. However, the energy gap is considered constant in thecollision processes; the dynamical effects on the band structureduring the collision are absent.

In this paper, we present the theoretical calculations ofthe SP values for protons and helium ions (He2+) traversingthe large band-gap insulators LiF (Eg=14 eV) and SiO2

(Eg=8 eV). In the simulations, we have the projectile ionshyperchanneled in the 〈001〉 axis of both materials; it meansthat the trajectory of the channeled ion is always far away fromthe atomic strings defining the channel. It is known that theelectronic energy loss can be favorably studied by utilizingchannel effect. Because in the hyperchanneling condition, theenergy transferred to the lattice atoms is considered to benegligible in comparison to electronic stopping, then stoppingof atomic projectiles traversing solids is completely dominatedby the electronic interaction processes. The behavior of thecalculated SP as a function of proton and helium ion velocityis discussed and compared with the experimental results. Byexploring the SP behavior in such a low-energy range we arenot aiming to obtain the SP values agreed quantitatively withthe experimental data, rather we are exploring the thresholdeffect of the SP, and trying to understand the underlyingphysics of these phenomena that we have observed.

The structure of the paper is as follows. In Sec. II we brieflyintroduce the theoretical framework and the computationaldetails of this work. In Sec. III we present the results relatedto three items. In the first one, the threshold velocities ofSP in LiF are presented and discussed. In the second partof this section, we analyze the threshold effects in SiO2, andthe results are discussed. In the third one, the electron transferhappened in the hyperchanneling condition and its effects on

the electronic energy loss are investigated. Conclusions arepresented in Sec. IV.

II. METHOD AND COMPUTATIONAL DETAILS

In the present work, in order to simulate the collisionprocess between ions and the target electrons, the Ehrenfest-TDDFT (E-TDDFT) model [23–26] is employed to run thecalculations. The framework consists of a quantum mechanicsdescription for the electrons and a classical approximation forthe ions. In this work, the calculations are performed by usingOCTOPUS ab initio code [27,28]. The Ehrenfest MD schemeis contained in the following coupled differential equations(atomic units are used here):

i∂�(x,t)

∂t= He(r, �R(t))�(x,t), (2)

MJ

d2 �RJ

dt2= −

∫dx�∗(x,t)[∇J He(r, �R(t))]�(x,t)

−∇J

∑I �=J

ZIZJ

| �RI (t) − �RJ (t)| , (3)

where �(x,t) is the many-body electron wave function.We define x ≡ {xj }Nj=1 and xj ≡ (�rj ,σj ) which includes thecoordinates �rj and the spin σj of the j th electron, and N is thenumber of electrons in the system. He(r, �R(t)) is the electronicHamiltonian and �R(t) ≡ { �R1(t),..., �RM (t)} is the instantaneousdistribution of the positions of all the nuclei; M is the numberof nuclei in the system. The motion of the nuclei is determinedby the set of Eq. (3). The electronic Hamiltonian is given by

He(r, �R(t)) = −N∑i

1

2∇2

i +∑i<j

1

|�ri − �rj |

−∑iJ

ZJ

| �RJ − �ri |. (4)

This form of the Hamiltonian allows one to write the forcethat acts on each nucleus in terms of the electronic densityn(�r,t); we can rewrite Eq. (3) as

MJ

d2 �RJ

dt2= −

∫d�rn(�r,t)[∇J He(r, �R(t))]

−∇J

∑I �=J

ZIZJ

| �RI (t) − �RJ (t)| . (5)

Equation (5) shows that the ionic force can be calculated fromthe time-dependent electronic density n(�r,t). This fact is thebasis of TDDFT-based Ehrenfest MD [29,30]. Instead of solv-ing Eq. (2), we can solve the corresponding time-dependentKohn-Sham (KS) system, which provides an approximation ton(�r,t)

i∂ϕi(�r,t)

∂t=

[−1

2∇2 + υKS[n](�r,t)

]ϕi(�r,t),

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

n(�r,t) = 2N/2∑i=1

|ϕi(�r,t)|2, (7)

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FIRST-PRINCIPLES STUDY OF THE THRESHOLD . . . PHYSICAL REVIEW A 89, 022707 (2014)

where ϕi(�r,t) is the Kohn-Sham electron orbital, υKS(�r,t) isthe time-dependent Kohn-Sham potential,

υKS(�r,t) = υext(�r,t) + υH (�r,t) + υxc(�r,t), (8)

with

υext(�r,t) = −∑

J

ZJ

| �RJ (t) − �r| , (9)

υH (�r,t) =∫

d�r ′ n(�r ′,t)|�r − �r ′| , (10)

where υH (�r,t) is the Hartree potential, υext(�r,t) and υxc(�r,t)are the time-dependent electron-nucleus and exchange-correlation potentials, respectively. In this model, the potentialenergy and forces acting upon the ions are calculated “onthe fly” as the simulation proceeds. This method allowsab initio MD (AIMD) simulation for excited electronic states.In Ehrenfest MD, transitions between electronic adiabaticstates are included, and it couples the populations of theadiabatic states to the nuclei trajectories [29]. It opens a wayto study the electron transfer between the ion and the targetelectrons during the collision [31].

In order to investigate electron excitation and electrondistribution after the collision, the electron distribution inthe conduction band is defined by the projection of thetime-dependent wave function to the initial particle states inthe conduction band [32,33],

nocc(m,k) =∑n′

|〈ϕmk|ϕn′k(T )〉|2, (11)

where m represents the KS orbital index and k is the Blochwave number.

In the present study, the interactions between the ioniccores and valence electrons are represented by the norm-conserving Troullier-Martins (TM) pseudopotential. An adia-batic exchange and correlation (XC) functional is employedin the time-evolving simulation, namely, the local densityapproximation (LDA) with Perdew-Wang analytic representa-tion [34]. The approximated enforced time reversal symmetrymethod [35] is employed to propagate the electronic wavefunctions, and the Verlet algorithm is used for the ionic motionequations. From numerical computation view, an excellentadvantage of combining TDDFT with Ehrenfest MD is that itprovides an orthogonalization free AIMD calculation, whichsaves a lot of computer resources and time.

In our calculations, the insulator crystalline films LiF andSiO2 are composed of 64 and 96 atoms, respectively. The ionchanneling processes are performed in a periodic simulationbox. A single k point (�) is used for integrations in theBrillouin zone. The external potential, electron density, andKS orbitals are discretized in a set of mesh grid points withuniform spacing of 0.18 A along all three spatial coordinatesin real space in the simulation cell. Before the time-dependentcalculations, all the valence electrons of the system are intheir ground states. The ground-state orbitals are calculated bydiagonalization of the time-independent KS Hamiltonian. Inthe real time evolution, the projectile ion core is initially placedoutside the simulation box. It starts moving along the center of〈001〉 channel (parallel to the z axis) with an initial velocity.

The time step for different incident velocities is adjusted withinthe range of about 10−3 fs to conserve the total energy.

In the time-dependent runs, the nuclei of the insulatoratoms are fixed in their equilibrium positions [1,21]. Thereare two reasons for treating them in such a way: (i) what weare concerned about is the energy obtained by the electronicsubsystem of the target. Fixing the atoms allows excluding thenuclear stopping from the energy loss. (ii) By fixing the sur-rounding lattice, the ionic positions and the electronic groundstate are made strictly periodic in space and the energy transfersirreversibly into the electronic subsystem. The correspondingelectronic SP is equal to the overall slope of the kinetic energyof the channeled ion with respect to its path length.

III. RESULTS AND DISCUSSION

A. Threshold effect in SP of LiF

The SP values of LiF for protons and He2+ ions as a functionof ion velocity are presented in Fig. 1. The calculated valuesand experimental data for SP of LiF for protons show the lineardependence on velocity [Fig. 1(a)]. For protons v < 0.8 a.u.,the calculated SP increases almost linearly with the projectilevelocity increasing, showing a velocity threshold at 0.1 a.u..At 0.9 a.u., the SP reaches the maximum value. For He2+ thereare some deviations from this linear dependence [Fig. 1(b)].This phenomenon is attributed to the electron transfer betweenHe2+ and LiF atoms in the stopping process, which is alsoconfirmed as an important electronic energy loss when theions are traversing solids.

Actually, our calculated SP values are much smaller than theexperimental data. In order to compare with the experimentaldata, we have to scale our SP results by a factor of 6 for

0.0 0.2 0.4 0.6 0.8 1.00

5

10

15

Velocity (a.u.)

−dE

/dx

(eV

/Å)

ref. [15] our data (scaled)

p → LiF

(b)

0

4

8

12 ref. [3]ref. [15]

ref. [20] ref. [21] our data (scaled)

He2+→LiF

(a)

FIG. 1. Electronic stopping power of LiF for (a) protons and(b) He2+ ions as a function of the incident velocity. Solid squares arethe scaled results of our calculations, and solid stars are calculatedresults from Ref. [21]. Also shown are the measured data fromRefs. [3] (solid circles), [15] (solid triangles), and [20] (soliddiamonds), respectively.

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FEI MAO, CHAO ZHANG, JINXIA DAI, AND FENG-SHOU ZHANG PHYSICAL REVIEW A 89, 022707 (2014)

LiF. There are methodical deficiencies [1,21] suggesting theobserved SP discrepancies between the calculated valuesand measured data that are inherent in the experimentalmeasurements. First, there is no perfect crystal structure inthe target films used in the experiments [36], which arefabricated by the evaporation process [3,15,20]. Second, theseexperimental values are averaged over random trajectories.Third, the densities of the films are very different from thoseof the bulk. The measured SP values are translated from theSCS (from Ref. [15]) by using the bulk atomic densities ofLiF and SiO2 for comparison. It is accepted that the ratioε of the channeled SP to the random SP (ε is defined as(−dE/dx)channel/(−dE/dx)random) should be at least by afactor of 0.5. However, ε < 0.5 is not prohibited. Examplesof ε < 0.5 were found for protons channeling in singlecrystal Germanium (Ge); ε<110>,<111> = 0.3 was measuredfor Ge [37,38]. It should be also stated that the measurementsof ε = 0.3 are for the hyperchanneling condition only [39].More discussions about the influences of channeling on the SPcan be found in the review [39].

Under our simulation conditions of hyperchanneling, itis difficult to obtain the SP values that agree well withthe experimental ones quantitatively. We have compared ourSP values of protons hyperchanneling in LiF〈001〉 with thesimulation results reported in literature [21], in which the SP isobtained in LiF〈011〉, and found that our SP values are lower bya factor of about 2. In order to clarify this discrepancy betweenthese two sets of theoretical results, we have carefully studythe detailed radial distribution of the electron density alongthe channel center of LiF〈001〉 and 〈011〉. We integrate theelectron density within the ion cylindrical track of variableradius r whose axis is the ion path. The electron density inthe track is then obtained by dividing the calculated value bythe volume of the track cylinder. The results are presentedin Fig. 2. The comparison of electron densities of the twochannels shows the electron density in the 〈011〉 channel islarger than that in the 〈001〉 channel by a factor of ∼2. Fora given velocity, according to the linear relationship between

0.0 0.5 1.0 1.5 2.00

10

20

30

40

Elec

tron

den

sity

(10-3

ele

ctro

ns/Å

3 )

Radius (Å)

LiF<001> LiF<011>

FIG. 2. The electron density inside a cylinder of variable radius r

whose axis is the center of LiF〈001〉 and 〈011〉 channels.

energy loss and electron density −dE/dx ∝n (n is the electrondensity), one can expect that the SP of 〈011〉 should be largerthan that of 〈001〉 by the same factor.

The hyperchanneling region is determined by the maximumclosed equipotential line around the channel axis. In this regionthe electron density is low compared with the rest of thecrystal. It can be seen from Fig. 2 that the electron densitydistribution in a given channel is not uniform. We have checkedthe SP dependence on the impact parameter when the ionsare traveling in the given channel. As the impact parameterdecreases, the electron density of the regions sampled bythe channeled ions increases quickly, the SP is significantlyenhanced, and then the agreement with the experimental datais greatly improved. One can see that the electron density inthe hyperchanneling region of LiF〈001〉 (when r � 0.3 A)is about 2.0 × 10−3 electrons/A3, but it increases to 1.2 ×10−2 electrons/A3 as it approaches the edges of the channel.We can scale our results from hyperchanneling to the casewhere all electronic density is probed by a factor of 6. Thenthe discrepancies between measured data and our results canbe successfully resolved. This implies that our results arequantitatively reasonable.

As seen in Fig. 1, the threshold velocities of proton andHe2+ in LiF are both at 0.1 a.u. which are in a good agreementwith the measured data [15]. In order to investigate thedistribution of the excited electrons after the collision, weperform the occupation analysis. From the time evolution ofthe KS spectra, the whole KS energy eigenvalues are disturbedby the channeling ion. For the collision between proton andLiF crystal at 0.1 and 1.0 a.u., the electron occupation numberdistribution in the conduction band are shown in Fig. 3. Theexcitation energy of the crystal has been extended to 10 eVin the conduction band. We can see clearly that the excitedelectrons are distributed broadly in the considered energyrange for the 1.0 a.u. proton. However, the electrons occupyonly a small part in the low excitation energies for the 0.1 a.u.proton, and the number of excited electrons in this energyrange is much smaller than that of the 1.0 a.u. proton. In the

0 2 4 6 8 100.00

0.02

0.04

0.06

0.08

0.10

Occ

up

atio

n (

per

un

it c

ell)

Energy (eV)

0.1 a.u.1.0 a.u.

FIG. 3. Occupation of excited electrons in the conduction bandof LiF after collision for 0.1 and 1.0 a.u. protons, respectively. Seemore details in the text.

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FIRST-PRINCIPLES STUDY OF THE THRESHOLD . . . PHYSICAL REVIEW A 89, 022707 (2014)

0.10 0.12 0.14 0.16 0.18 0.200.00

0.01

0.02

0.03

0.04

0.05

dE

/dx

( eV

/Å)

Spacing (Å)

FIG. 4. The electronic stopping power of LiF as a function ofthe spacing ranging from 0.1 to 2.0 A for the 0.1 a.u. protonhyperchanneling in the LiF〈001〉 direction.

realistic system, electrons not excited to the conduction bandare suppressed by the energy gap. This means the projectileion does not lose any energy to the target electron subsystemwhen the velocities are below a certain threshold, thus resultingin a threshold effect in the SP dependence on the velocity. Asthe projection calculations are very computational demanding,the target LiF crystal is composed by 32 atoms in this kind ofcalculation. Although the empty KS level does not provide theexact description of the real excited states of the nanostructure,this can be considered as the first approximation to the trueexcitations of the system [40–42].

In order to confirm the validity of the threshold effect,we have checked the convergence of threshold effect withrespect to the spacing of the mesh grid points in real space.The OCTOPUS code works without a basis set, in which the KSorbitals are numerically represented by the mesh gird points.The accuracy of the simulations is controlled by the separationbetween points in real space, or spacing. We have checkedthe convergence of the threshold effect by reducing the gridspacing when the 0.1 a.u. proton is traversing the center ofLiF〈001〉, and the results are shown in Fig. 4. As shown in thisfigure, the SP is almost zero when the spacing ranges from0.1 to 2.0 A. We can conclude that the threshold effect doesnot depend on the magnitude of the spacing and it is wellconverged.

B. Threshold effect in SP of SiO2

The electronic SP of SiO2 for proton is shown in Fig. 5(a).The velocity-proportional SP is well produced. It indicatesthat the threshold velocity is at 0.1 a.u., which is in goodagreement with the results reported in the literature [15].However, there are deviations between the calculated valuesand the experimental ones which are also obtained from thebulk atomic density of SiO2. The agreement is achieved whenthe calculated results are scaled by a factor of 5. For the caseof oxide SiO2, besides the anomalous low electron density inthe hyperchanneling condition accounting for the very low

0

10

20

30

0.0 0.2 0.4 0.6 0.8 1.00

5

10

15

20

−dE

/dx

(eV

/Å)

ref. [15] our data (scaled)

p → SiO2

(a)

Velocity (a.u.)

ref. [15] our data (scaled)

He2+→ SiO

2(b)

FIG. 5. Electronic stopping power of SiO2 for (a) protons and(b) He2+ ions as a function of the incident velocity. Solid squaresare the scaled calculation results, while solid triangles are themeasurement values [15]. Dashed line is the result of linear fitting tothe experimental data.

SP, another reason may come from the strong binding ofoxide. The strong chemical effect is significantly decreased inthe hyperchanneling condition in comparison with that in therandom condition, which would cause a large deviation in SPat low energies [36]. Figure 5(b) shows the SP results (scaledby 2) for He2+, compared to the SP behavior of LiF for thesame ion; the theoretical SP of SiO2 is nicely proportional tovelocity. Considering the velocity-proportionality assumption,we extrapolate the experimental SP data down to zero velocity;that results in a zero value for the SP, which means that thereis no threshold velocity for SiO2 stopping helium ions [15].

In our simulations, the linear behavior SP for He2+ ionsimplies that the cutoff value is at 0.2 a.u.. Actually, the He2+is slowed down and loses energy only in the initial part of thehyperchanneling path at 0.1 and 0.2 a.u.. In the following path,the ions do not lose any energy. The force acting on heliumions in the z axis direction for various velocities is presentedin Fig. 6(a). The projectile moves from 6.7 to −6.7 A in thecrystalline channel. For 0.2 a.u. He2+, after traveling about 6atom layers (at ∼2 A), the applied force on He2+ varies like asine function with respect to the displacement, and its directionturns up and down periodically (in the z axis direction), leadingto no net energy loss in the following path. It moves like aharmonic oscillator, which reflects the periodic properties ofthe crystalline lattice structure.

When the incident energy increases, the ions are exposedto purely drag force along the channel. As shown in Fig. 6(a),with increasing the incident velocity, the locations of peaksin the force curves move from the gaps between the O andSi atoms layers to the next O atomic layers. This is resultedfrom the maximum electron density experienced by the ionswhen they are passing through these O atom layers. This leadsto such an interesting motion scenario of channeled He2+ ions.

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FEI MAO, CHAO ZHANG, JINXIA DAI, AND FENG-SHOU ZHANG PHYSICAL REVIEW A 89, 022707 (2014)

-8 -6 -4 -2 0 2 4 6 8

0

20

40

60

80

100

120

-5

0

5

10

15

20

25

30

35

En

ergy

(eV

)

(a)

(b)

For

ce (

eV/Å

)

0.2 a.u. 0.4 a.u. 0.6 a.u. 1.0 a.u. O layers Si layers

Displacement (Å)

FIG. 6. (Color online) (a) The applied force and (b) the kineticenergy of He2+ ions hyperchanneling along the SiO2〈001〉 directionas a function of displacement for several velocities. The vertical blacksolid lines are indications of the positions of silicon atom layers, whilethe vertical red dashed lines are for oxygen atom layers. See moredetails in the text.

When it moves from the Si layers to O layers, the resistantforce is getting bigger, thus giving rise to a primary energyloss in these intervals during the hyperchanneling path. Whilethe He ions move from the O layers to Si layers, the resistanceis getting smaller, generating a plateau region in the energyloss curves during these intervals. However, this mechanismis energy dependent. The traversing ion always loses energyalong its channel path in the high-energy region, such as at1.0 a.u..

From the analysis of the He2+ ion kinetic energy behaviorwith respect to the displacement at 0.2 a.u., it can be seenfrom Fig. 6(b) that the ion doesn’t lose any energy afterA (the kinetic energy curves are shifted down to ensure thefinal state energy values are at zero in this figure), which isconsistent with the force situation. In order to explain thispuzzling behavior of the energy loss, we have analyzed thecharge state of the hyperchanneling ions. It is found that theincident He2+ is neutralized when it just passes the outmostatomic layers of the film in the case of 0.1 and 0.2 a.u., whichmeans that the ion reaches the equilibrium charge state at thebeginning of its passage through the target. This results in theoscillatory behavior of the applied force, and no energy lossof the ion in the following hyperchanneling path. As impliedin the literature [5], in the very low-energy range, the repeatedionization processes of helium ion traversing SiO2 lead to thecontinuous energy loss of the projectile, and this is the reasonthat no threshold velocity is found in He ions transmitting SiO2

in the experiments. However, the recurring charge exchange

processes between He ions and SiO2 cannot be captured bythe present model employed in our simulations.

The magnitude difference of SP between LiF and SiO2 isrelated to their different lattice and electronic structure. In thecase of LiF and SiO2, there are eight and 16 valence electronsper molecule in the pseudopotential model, respectively. Thelattice parameter of LiF (a = 4.0 A) is bigger than that of SiO2

(a = 2.5 A, the side length of the square channel); it results in amuch higher electron density of SiO2 in comparison with thatof LiF. Moreover, from the electronic density contour obtainedfrom the ground-state calculations, the electronic density ofSiO2 is distributed more uniformly in space. This feature isvery important for the material effectively stopping the movingions. Because when the ion is under the hyperchannelingcondition in SiO2, it is traversing the electron cloud along itspath, unlike the situation when the ions are hyperchannelingin LiF. These factors lead to the SP values of proton and He2+in SiO2 being higher than those in LiF crystal.

C. Electron transfer in hyperchanneling condition

In order to demonstrate the electron transfer process, theelectron density evolution of the whole system when the protonis hyperchanneling in LiF〈001〉 is shown in Fig. 7. Before theion approaches the crystal, it is a bare ion. When it collideswith the electrons of the lattice atom, the ion captures someelectrons. From the time evolution of the electron density, wecan find that the leaving ions retain some captured electronsafter collision. The number of transferred electrons is obtained

FIG. 7. (Color online) Time evolution of three-dimensional elec-tron density of proton in LiF 〈001〉 hyperchanneling geometry at0.1 a.u. (a) t=2.72 fs; the proton is above the crystal. (b) t = 5.45 fsand (c) t = 9.0 fs; the ion is penetrating the channel and gets electronsfrom the F− lattice site. (d) t=10.9 fs; the proton retains someelectrons after the collision. See more details in the text.

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FIRST-PRINCIPLES STUDY OF THE THRESHOLD . . . PHYSICAL REVIEW A 89, 022707 (2014)

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

2.5

3.0

Tra

nsf

erre

d c

har

ge(u

nits

of e- )

Velocity (a.u.)

He ion in LiF He ion in SiO2

FIG. 8. The number of electrons captured by He2+ hyperchan-neled in the 〈001〉 direction of SiO2 and LiF as a function of velocity.See more details in the text.

by integrating the valence charge density around the ionnucleus within a radius of 2.0 A. The results for He2+ areshown as a function of the velocity in Fig. 8. Compared tothe case of LiF, one can find that the He2+ captures moreelectrons in SiO2, and the number of transferred electrons isless sensitive to the incident velocity. It implies that the energyloss due to electron transfer is almost constant for SiO2 in theconsidered energy range. The electronic excitation leads to thelinear dependence, and the contribution of the electron transferjust shifts the SP values uniformly, so that the SP velocitydependence is still linear. This is essentially important for theSP dependence on the velocity (v � 0.2 a.u.). The situationis different from that observed in LiF. In the case of LiF,the number of captured electrons decreases as the velocityincreases.

The different quantities of the transferred electrons areattributed to the bond characteristics of LiF and SiO2. LiFis the ionic crystal insulator, in which the electrons are allconcentrated on the F− lattice site, and the valence bandformed by 2p electrons of F− is located below the Fermienergy by 14 eV. In the case of SiO2, most of the valenceelectrons are distributed in the interstitial regions betweenthe adjacent atoms. The strength of the ionic bond is muchhigher than that of the covalent compound SiO2. Whenthe projectile passes through the center of the channel, thedistance between the ion path and the F− (about 2.8 A)in LiF is bigger than that (about 1.76 A) in SiO2; itaccounts for the He2+ ions capturing less electrons from thesurrounding F− sites. These features result in relatively largeelectron capture cross sections for the ion channeled in SiO2.

IV. CONCLUSIONS

In conclusion, by employing the TDDFT-MD method, wehave studied the electronic SP of LiF and SiO2 for very low-energy protons and He2+ ions hyperchanneling in the 〈001〉axis of these materials. The velocity-proportional electronic SPof LiF and SiO2 is predicted. A quantitative agreement betweenthe experimental data and our results is achieved when theresults obtained from hyperchanneling are scaled to the casewhere all electronic density is probed. We have reproducedthe measured velocity thresholds at 0.1 a.u. for protons andHe2+ ions in LiF, and for protons in SiO2. The validity ofthe threshold effect is confirmed by the convergence test. Wehave investigated the distribution of the excited electrons inLiF after collision. The number of valence electrons which areexcited to the conduction band is very small at 0.1 a.u.. Thisdoes not result in effective electronic energy loss.

The linear dependence of the SP for He2+ ions in SiO2

reveals the threshold velocity is at 0.2 a.u.. It is explainedby the early neutralization of He2+ when hyperchanneled inSiO2. We have analyzed the effects of the charge transfer onthe SP values. Because of the covalent bond characteristicof SiO2, the electrons are distributed in the channel region.The resistance force felt by the ions in SiO2 is larger thanthat in LiF, which leads to much bigger SP values of SiO2. Ourcalculations reveal that the electronic SP is related to the chargestate, mass, and kinetic energy of the projectile. Moreover, itis extremely dependent on the lattice and electronic structureof the target, such as band gap and inhomogeneities of theelectron distribution in solids.

In this study, LDA is employed for approximation of the XCterm in the energy functional. LDA works well for the systemswith very small electron density gradients in space and time. Itdoes not give an accurate description for the electronic systemlike the ionic compound LiF, in which the electron densitychanges drastically. As revealed in this study, the electrondistribution is critically important for stopping ions when theyare traversing the solids. What’s more, the LDA approximatesthe XC functional as being local in time and space. In principle,because of the locality nature of TDLDA, it does not givean accurate description for the electron transfer behavior inthe collision processes. Our results should not be treated asquantitative predictions of the charge transfer.

ACKNOWLEDGMENTS

The authors thank F. Wang for fruitful discussions.This work was supported by the National Natural ScienceFoundation of China under Grants No. 11025524 and No.11161130520, National Basic Research Program of Chinaunder Grant No. 2010CB832903, and the European Com-mission’s 7th Framework Programme (FP7-PEOPLE-2010-IRSES) under Grant Agreement Project No. 269131.

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