PHYSICAL REVIEW B 15 OCTOBER 1998-IIVOLUME 58, NUMBER 16

Electronic stopping power of aluminum crystal

I. Campillo and J. M. PitarkeMateria Kondentsatuaren Fisika Saila, Zientzi Fakultatea, Euskal Herriko Unibertsitatea, 644 Posta kutxatila,

48080 Bilbo, Basque Country, Spain

A. G. EguiluzDepartment of Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee 37996-1200

and Solid State Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831-6032~Received 1 April 1998!

Ab initio calculations of the electronic energy loss of ions moving in aluminum crystal are presented, withinlinear-response theory, from a realistic description of the one-electron band structure and a full treatment of thedynamical electronic response of valence electrons. For the evaluation of the density-response function we usethe random-phase approximation and, also, a time-dependent extension of local-density-functional theory. Weevaluate both position-dependent and random stopping powers, for a wide range of projectile velocities. Ourresults indicate that at low velocities band-structure effects slightly enhance the stopping power. At velocitiesjust above the threshold velocity for plasmon excitation, the stopping power of the real solid is found to besmaller than that of jellium electrons, corrections being of about 10%. This reduction can be understood fromsum rule arguments.@S0163-1829~98!08339-8#

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I. INTRODUCTION

The stopping power for charged particles penetratinsolid has been the topic of considerable theoretical andperimental interest, since the beginning of this century1–4

The electronic stopping power due to collisions with valenelectrons has been evaluated for many years on the basisjellium model5,6 of the target, the electronic states being dscribed by plane waves. However, in a more realisticproach valence electrons move in a periodic potential, etronic states are described by Bloch wave functions, andspectrum of one-electron excitations splits into the so-caenergy bands. The impact of band structure effects on bplasmon dispersion curves7 and dynamical structurefactors8–10 has been investigated recently, demonstratingimportance of these effects even in the case of free-electlike metals such as aluminum.

Among the most recent attempts to fully introduce telectronic band structure in the evaluation of the stopppower for low projectile velocities there is, for alkaline meals, a one-band calculation,11 as well as a calculation baseon a linear combination of atomic orbitals.12 For arbitraryincident velocities the stopping power can be calculatwithin linear-response theory, from the knowledge of the dnamical density-response function of the target.13–15 Ap-proximate semiempirical treatments of this quantity habeen made, and stopping powers of silicon16–19 and gold20

for channeled ions have been predicted. More recently,low velocity limit has been investigated, on the basis ostatic treatment of the density response.21 First-principlestreatments of the full dynamical electronic response of vous solids have also been performed, and preliminaryabinitio evaluations of the stopping power of real solids habeen presented.22–24

In this paper we investigate, within linear-respontheory, the valence electronic energy loss of ions mov

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through aluminum crystal. Aluminum is well known to betypical jelliumlike metal with a well-defined excitation spetrum, and it represents, therefore, an appropriate benchmfor ab initio evaluations of the electronic stopping powerreal solids. First of all, in Sec. II we describe our full treament of the wave-vector and frequency-dependent electrresponse of valence electrons, based on a realistic descriof the one-electron band structure and first-princippseudopotential theory. For the evaluation of the densresponse function we use the random-phase approximati25

~RPA! and, also, a time-dependent extension of local-denfunctional theory~TDLDA !.26 In Sec. III, we derive explicitexpressions for both random and position-dependent sping powers, from the knowledge of the imaginary partthe projectile self-energy, which is evaluated in the so-calGW approximation.27 In Sec. IV, numerical calculations othe stopping power of valence electrons in aluminum crysare presented, for a wide range of projectile velocities, awe compare our results with the stopping power of a homgeneous electron gas with a density equal to that of alunum. We evaluate, separately, contributions to the eneloss coming from the excitation of single electron-hole paand plasmons, and we interpret our results on the basis of -sum-rule for the dynamical structure factor. We only cosider the electronic response of valence electrons, andcontribution to the electronic stopping power coming frothe excitation of core electrons is, therefore, not taken iaccount. In Sec. V, our conclusions are presented.

II. DENSITY-RESPONSE FUNCTION

The linear density-response functionx(r ,r 8,v) of anelectron system is defined by the equation

r ind~r ,v!5E d3r 8x~r ,r 8,v!Vext~r 8,v!, ~2.1!

10 307 © 1998 The American Physical Society

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10 308 PRB 58I. CAMPILLO, J. M. PITARKE, AND A. G. EGUILUZ

wherer ind(r ,v) is the electron density induced by an extenal potentialVext(r ,v).

In a self-consistent-fieldtheory, the induced electron density is derived from the response functionx0(r ,r 8,v) fornoninteracting electrons moving in an effective potenVe f f(r ,v), as follows:

r ind~r ,v!5E d3r 8x0~r ,r 8,v!@Vext~r 8,v!1Vind~r 8,v!#,

~2.2!

whereVind(r ,v) represents the linear change inVe f f(r ,v)brought about by the induced electron density itself. Sinwe consider a time-dependent external field, we are inmore general scenario of time-dependent density-functiotheory ~DFT!,28 the theorems of which29 generalize those othe usual DFT.30

Within time-dependent DFT,Vind(r ,v) consists of thesum of two terms~we use atomic units throughout, i.e.,me5e5\51!: The Hartree contribution,

VHind~r ,v!5E d3r 8

1

ur2r 8ur ind~r 8,v!, ~2.3!

which accounts for the average~long-range! effects of theCoulomb interaction between the target electrons, andexchange-correlation~XC! contribution,

Vxcind~r ,v!5E d3r 8

dVxc@r#

dr~r ,v!r ind~r 8,v!, ~2.4!

which accounts for the effects of all many-body short-rancorrelations not included in the Hartree approximation. HeVxc represents the functional derivative of the XC enerfunctionalExc :

Vxc@r#5dExc@r#

dr~r ,v!. ~2.5!

Introduction of Eqs.~2.3! and ~2.4! into Eq. ~2.2! andcomparison with Eq.~2.1! result in the following integralequation for the self-consistent-field density-response fution:

x~r ,r 8;v!5x0~r ,r 8;v!1E dr1E dr2x0~r ,r1 ;v!

3V~r1 ,r2 ;v!x~r2 ,r 8;v!, ~2.6!

where

V~r ,r 8;v!51

ur2r 8u1Kxc@r# ~2.7!

and

Kxc@r#5d2Exc@r#

dr~r ,v!dr~r 8,v!. ~2.8!

In the time-dependent Hartree or random-phase apprmation,Ve f f(r ) consists only of the average electrostaticteraction between the electrons, the induced potential isgiven by the Hartree term exclusively, andKxc(r ,r 8) in Eq.

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~2.7! is taken to be zero. In the TDLDA, the kernelKxc(r ,r 8)entering Eq.~2.7! is replaced by

Kxc~r ,r 8!LDA5d~r2r 8!FdVxc~r!

dr Gr5r0~r !

, ~2.9!

whereVxc(r) is the derivative of the XC energy of a homogeneous electron gas of densityr, which we compute withuse of the Perdew and Zunger parametrization.31 This so-called TDLDA represents an adiabatic extension~a zero fre-quencyKxc is used! to finite frequencies of the local-densitapproximation~LDA ! for XC.

For periodic crystals we introduce a Fourier expansionthe density-response function,

x~r ,r 8,v!51

V (q

BZ

(G,G8

ei ~q1G!•re2 i ~q1G8!•r8xG,G8~q,v!,

~2.10!

whereV represents the normalization volume, the first suruns overq vectors within the first Brillouin zone~BZ!, andG andG8 are reciprocal lattice vectors. Then, introductionEq. ~2.10! into Eq. ~2.6! leads to the following matrix equation:

xG,G8~q,v!5xG,G80

~q,v!1(G9

(G-

xG,G90

~q,v!

3VG9,G-~q,v!xG-,G8~q,v!, ~2.11!

whereVG9,G-(q,v) represent Fourier coefficients of the interaction potential of Eq.~2.7!. The Fourier coefficients ofthe density-response function of noninteracting electrhave the well-known form32

xG,G80

~q,v!51

V (k

BZ

(n,n8

f k,n2 f k1q,n8Ek,n2Ek1q,n81\~v1 ih!

3^fk,nue2 i~q1G!•rufk1q,n8&

3^fk1q,n8uei~q1G8!•rufk,n&, ~2.12!

where the second sum runs over the band structure for ewave vectork in the first BZ, f k,n are Fermi factors, andfk,n(r ,v) are single-particle Bloch states.

In the RPA, the one-electron Bloch states entering~2.12! are the self-consistent eigenfunctions of the onelectron Hartree Hamiltonian, and the coefficienVG9,G-(q,v) entering Eq.~2.11! are the Fourier coefficientsof the bare Coulomb interaction. In the TDLDA, the onelectron Bloch states entering Eq.~2.12! are the self-consistent LDA solutions of the Kohn-Sham equationDFT, and the coefficientsVG9,G-(q,v) entering Eq.~2.11!are the Fourier coefficients of the interaction potential of E~2.7! with the XC kernel of Eq.~2.9!.

For the evaluation of the one-electron Bloch states,first expand them in a plane-wave basis,

fk,n~r !51

AV(G

uk,n~G!ei ~k1G!•r, ~2.13!

with a kinetic-energy cutoff of 12 Ry, which correspondskeeping approximately 100G vectors in Eq.~2.13!. Then we

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PRB 58 10 309ELECTRONIC STOPPING POWER OF ALUMINUM CRYSTAL

solve for the coefficientsuk,n self-consistently, within a fulldescription of the electron-ion interaction based on the usa nonlocal, norm-conserving ionic pseudopotential.33 Wesubsequently evaluate, from Eq.~2.12!, the Fourier coeffi-cientsxG,G8

0 (q,v), which is the most demanding part of thresponse calculation. Finally, we solve the matrix equat~2.11! for the Fourier coefficients of the interacting responfunction xG,G8(q,v), which we obtain in both RPA andTDLDA.

III. ELECTRONIC STOPPING POWER

We consider a particle of chargeZ1 moving in an inho-mogeneous medium. The damping rate of the projectilethe statef0(r ) with energyE0 is obtained from the knowl-edge of the imaginary part of the self-energyS(r ,r 8;E0),according to

g522E drE dr 8f0* ~r !Im S~r ,r 8;E0!f0~r 8!. ~3.1!

In the so-calledGW approximation, the self-energy igiven by27

S~r ,r 8;v!5i

2p E2`

`

dv8G~r ,r 8;v2v8!W~r ,r 8;v8!,

~3.2!

where G(r ,r 8;v) represents the Green’s function for thprojectile, andW(r ,r 8,v) is the screened interaction:

W~r ,r 8;v!5v~r2r 8!1Z12E dr1E dr2v~r2r1!

3x~r1 ,r2 ,v!v~r22r 8!, ~3.3!

v(r2r 8) being the bare Coulomb interaction andx(r ,r 8,v)the density-response function of the medium. For periocrystals, we introduce the Fourier representation givenEq. ~2.10! into Eq. ~3.3!, and we find that

W~r ,r 8,v!5Z1

2

V (q

BZ

(G,G8

ei ~q1G!•re2 i ~q1G8!•r8

3vG~q!@dG,G81xG,G8~q,v!vG8~q!#,

~3.4!

where vG(q)54p/uq1Gu2 are the Fourier coefficients othe bare Coulomb interaction.

Replacing the Green’s function entering Eq.~3.2! by itszeroth-order approximation, we find the following expresion for the damping rate:

g5E0

`

dvPv , ~3.5!

where

Pv522(fE drE dr 8f f* ~r 8!f0* ~r !Im W~r ,r 8,E02Ef !

3f f~r !f0~r 8!d@v2~E02Ef !#, ~3.6!

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the sum running over a complete set of final statesf f(r ) ofenergyEf .

The quantityPv entering Eq.~3.5! represents the probability for the projectile to transfer energyv to the medium.Consequently, the stopping power of the medium, i.e.,energy loss per unit path length of the projectile, canobtained as follows:

2dE

dx5

1

v E0

`

dvvPv , ~3.7!

wherePv is given by Eq.~3.6!.

A. Random stopping power

In the case of heavy projectiles moving at a constantlocity v with no definite impact parameter, the initial anfinal states can be described by plane waves:

f0~r !51

AVeiq0•r ~3.8!

and

f f~r !51

AVei ~q02q!•r, ~3.9!

whereq0 and q represent the initial momentum of the projectile and the momentum transfer, respectively.

For heavy projectiles recoil can be neglected, i.e.,

E02Ef5q•v, ~3.10!

and after introduction of Eqs.~3.4!, ~3.8!, ~3.9!, and ~3.10!into Eq. ~3.6! we find the following expression for the socalled random stopping power:

F2dE

dxGrandom

528p

VvZ1

2(q

BZ

(G

~q1G!•v

uq1Gu2

3Im eG,G21 @q,~q1G!•v#, ~3.11!

whereeG,G821 (q,v) is the inverse dielectric matrix in momen

tum space:

eG,G821

~q,v!5dG,G814p

uq1Gu2 xG,G8~q,v!, ~3.12!

the density-response matrixxG,G8(q,v) being given by Eq.~2.11!.

The symmetry of the one-particle Bloch states resultsthe following identity:

eG,G821

~Sq,v!5eS21G,S21G8~q,v!, ~3.13!

with S representing a point group symmetry operation inperiodic crystal. As a consequence, the stopping poweEq. ~3.11! can be evaluated from the knowledge of theelectric matrix for wave vectorsq lying in the irreducibleelement of the Brillouin zone~IBZ!:

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10 310 PRB 58I. CAMPILLO, J. M. PITARKE, AND A. G. EGUILUZ

F2dE

dxGrandom

528p

VvZ1

2(q

IBZ

(S

(G

~Sq1G!•v

uSq1Gu2

3Im eS21G,S21G21

3@q,~Sq1G!•v#, ~3.14!

with the second sum in this equation running over the symetry operations generating the wave vectors in the staeachq.

If the diagonal elements of the inverse dielectric matentering Eq.~3.11! are replaced by the inverse dielectrfunction of a homogeneous electron gas,5 i.e.,

eG,G21 ~q,v!→ehomog

21 ~ uq1Gu,v!. ~3.15!

Equation~3.11! exactly coincides with the well-known formula for the stopping power of a homogeneous electgas.34

B. Position-dependent stopping power

In the case of heavy projectiles moving with constantlocity v on a definite trajectory at a given impact parameb, the initial and final states can be described in termsplane waves in the direction of motion and ad function inthe transverse direction.35 Then, introduction of these statef0(r ) and f f(r ), and the screened interaction of Eq.~3.4!into Eq.~3.6! gives, after neglecting the projectile recoil, thfollowing result for the position-dependent stopping powe

F2dE

dx Gb

528p

VvZ1

2(q

IBZ

(S

(G

(K

8

3eiK•b~Sq1G!•v

uSq1G1K u2Im eS21G,S21~G1K !

21

3@q,~Sq1G!•v#, ~3.16!

the sum(K8 being restricted to those reciprocal lattice vetors that are perpendicular to the velocity of the projecti.e., K•v50. eG,G8

21 (q,v) is the inverse dielectric matrix oEq. ~3.12! and S represents, as in Eq.~3.14!, a symmetryoperation of the point group of the periodic crystal.

The most important contribution to the positiodependent stopping power of Eq.~3.16! is provided by theterm K50, the magnitude of the other terms dependingthe direction of the velocity. For those directions for whithe condition ofK•v50 is never satisfied we have the radom stopping power of Eq.~3.14!, and for a few highlysymmetric orchannelingdirections non-negligible corrections to the random result are found, thus exhibiting theisotropy of the position-dependent stopping power. We anote that ~a! the average over impact parameters ofposition-dependent stopping power of Eq.~3.16! along anygiven channel has the same value as the random stoppower of Eq.~3.14!, and~b! as long as the diagonal elemenof the so-called dynamical structure facto22 Im xG,G(q,v), were isotropic, there would be no depedence of the random stopping power on the direction ofv.

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IV. RESULTS AND DISCUSSION

The input of our calculation of both random and positiodependent stopping powers of Eqs.~3.14! and ~3.16! is theinteracting response matrixxG,G8(q,v) of Eq. ~2.11!, whichwe have solved for Al crystal. The number of bands requiin the evaluation of the polarizabilityxG,G8

0 (q,v) of Eq.~2.12! depends on the value of the frequencyv. As the maxi-mum energyv transferred by the moving projectile to thtarget is proportional to the velocity, only a few bands arequired in the case of slow projectiles. Indeed, it has bfound22 that at velocities belowv50.1 a.u. just two bandsare required to account for the energy loss. At larger veloties excitations to higher bands become possible, and wethat convergence is achieved, forv,2 a.u., if one considerson the order of 30 bands. The results presented below hbeen found to be well converged for all velocities undstudy, and they all have been performed with the use ofbands.

The sampling of the BZ required for the evaluationboth the polarizability of Eq.~2.12! and the stopping powerof Eqs. ~3.14! and ~3.16! has been performed on 10310310 Monkhorst-Pack meshes36 ~47 points in the IBZ! withan energy smearing of the Fermi level of 0.25 eV, and tsampling has been done in conjunction with a choice ofinite value of the damping factor entering Eq.~2.12! of h51.5 eV. At velocities of the projectile below the velocitfor which plasmon excitation becomes possible, convergeis already achieved with the use of 43434 Monkhorst-Packmeshes~eight points in the IBZ!.

A. Random stopping power

The main ingredient in the calculation of the random stoping power of Eq.~3.14! is the so-called dynamical structurfactorSG(q,v). It is proportional to the diagonal elementsthe imaginary part of the density-response matrix:

S~q1G,v!522 Im xG,G~q,v!

52uq1Gu2

2pIm eG,G

21 ~q,v!, ~4.1!

whereq is taken to be in the first BZ. Thus, couplings of thwave vectorq1G to wave vectorsq1G8 with GÞG8,which appear as a consequence of the existence of elecdensity variations in real solids, only contribute to the radom stopping power through the dependence of the diagelements of the interacting response matrixxG,G(q,v) onthe off-diagonal elements of the polarizabilityxG,G8

0 (q,v).We have found that in the case of Al crystal, which doespresent strong electron density gradients nor special elecdensity directions~bondings!, contributions from these socalled crystalline local-field effects are within 0.5% of thtotal random stopping power. Thus, as for the evaluationthe random stopping power of Al crystal, the off-diagonelements ofxG,G8

0 (q,v) can be neglected.In the general case of an inhomogeneous system

finds, for the imaginary part of the inverse dielectric funtion, the following form of thef -sum rule:

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PRB 58 10 311ELECTRONIC STOPPING POWER OF ALUMINUM CRYSTAL

E0

`

dvv Im eG,G821

~q,v!522p2nG2G8 , ~4.2!

wherenG represent the Fourier components of the densNote that nG50 , which equals the average density of tsystem, does not depend on the details of the band strucThus, it may be argued that band-structure effects, whhave an impact on both plasmon dispersion7 and the dynami-cal structure factor,10 may give no corrections to integratequantities as the random stopping power. However, becain the sum over the wave vectorq in Eq. ~3.14! the fre-quencyv in the argument of the structure factor takes valufrom 0 to only uq1Guv, the stopping power of band electrons and that of electrons in an effective jellium with tsame average electron density will be, in general, differe

In Fig. 1 we show, as a function of the velocity of thprojectile, our full calculation of the random stopping pow@see Eq.~3.14!# of valence electrons in Al crystal for proton(Z151), together with the corresponding result for a homgeneous electron gas with an electron density paramequal to that of aluminum (r s52.07).37 In the calculation ofthe stopping power of both band and jellium electrons,third sum in Eq.~3.14! is extended over 15G vectors of thereciprocal lattice, the magnitude of the maximum momenttransferq1G being 2.9qF ~qF is the Fermi momentum!.38,39

Solid and open circles represent our calculated random sping powers of band electrons, as calculated within the Rand the TDLDA, respectively. Solid and dashed lines repsent the corresponding stopping powers of jellium electroAll of these results have been found, for Al crystal, toinsensitive to the choice of the projectile velocity directio

In the case of projectiles moving at low velocities, belothe threshold velocity for plasmon excitation, the sum othe frequencyv5(q1G)•v in Eq. ~3.14! can never be re-placed by an integration from 0 to as in Eq.~4.2!, and thestopping power will, in principle, depend on the band struture of the crystal. The existence of interband transitions,present within a jellium model of the crystal, result indynamical structure factor which is, at low frequenci~where it increases linearly with frequency!, slightly en-

FIG. 1. Full band-structure calculation, from Eq.~3.14!, of therandom stopping power of valence electrons in Al crystal for ptons (Z151), as a function of the projectile velocity. Solid anopen circles represent the results obtained in the RPA andTDLDA, respectively. RPA and TDLDA stopping powers of eletrons in an homogeneous electron gas withr s52.07 are representeby solid and dashed lines, respectively.

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hanced with respect to the corresponding jellium result.10 Asa result, the stopping power of the real target is, for projecvelocities smaller than the Fermi velocity and within boRPA and TDLDA, a linear function of the velocity and abo7% higher than the stopping power of jellium electrons.

The threshold velocity for which plasmon excitation bcomes possible isv t'1.3 a.u. in the case of electronsjellium with r s52.07. In the case of Al crystal this threshovelocity is slightly smaller (v'1.2 a.u.), as can be easilconcluded from an inspection of the plasmon disperscurve for Al crystal of Ref. 7. For projectile velocities largethan v t there are two mechanisms of valence electron extation: ~a! excitation of single electron-hole pairs and~b!excitation of plasmons. Figure 2 exhibits, as a function ofprojectile velocity, separate contributions to the stopppower of both band~solid circles! and jellium ~solid lines!electrons coming from electron-hole pair excitation@Fig.2~a!# and plasmon excitation@Fig. 2~b!#. Although for wavevectors that are smaller thanqc ~the critical wave vectorwhere the plasmon dispersion enters the electron-holeexcitation spectrum! both mechanisms of valence electroexcitation contribute to the energy loss, contributions frolosses to electron-hole pair excitations are negligible foq,qc , and Eq.~4.2! leads us to the conclusion that contribtions from losses to plasmon excitation are independenthe detailed band structure of the crystal. That this is the cis obvious from Fig. 2~b!. As for the contribution to the

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FIG. 2. Contributions to the random stopping power of valenelectrons in Al crystal coming from the excitation of~a! electron-hole pairs and~b! plasmons, versus the projectile velocity, as otained from Eq.~3.14! within the RPA. Full band structure calculations are represented by solid circles. The corresponding jellresults are represented by solid lines.

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10 312 PRB 58I. CAMPILLO, J. M. PITARKE, AND A. G. EGUILUZ

energy loss coming from the excitation of electron-hopairs, Figs. 1 and 2~a! show that band-structure correctionlower the stopping power of electrons in jellium by abo10% at and just above the plasmon threshold velocity; thia consequence of the dynamical structure factor for thecrystal being, within the electron-hole pairs continuum aq.qc and v,(q1G)v, smaller than it is in the case ofhomogeneous electron gas. Nevertheless, contributions tstopping power coming from the excitation of plasmonsstill smaller than contributions from losses to electron-hpairs, and there is, at high velocities, exact equipartitionthe energy loss, as in the case of a homogeneous elegas.40 Also, at high velocities the sum over the frequenv5(q1G)•v in Eq. ~3.14! can be replaced by an integratiofrom 0 to` and the stopping power of both jellium and baelectrons is expected to be the same, as illustrated in Figand 2~a!.

We note that the TDLDA results plotted in Fig. 1 for thstopping power of jellium electrons, which describe shorange correlation effects within the LDA, approximately rproduce more detailed calculations with either static ornamic local-field corrections included.41 The stopping powerof both jellium and band electrons is enhanced by ab20%, at low velocities~exclusive electron-hole pairs domain!, with respect to the RPA result, as a consequenceshort-range correlation effects provoking a reduction inscreening within the target. The sum rule of Eq.~4.2! equallyapplies for both RPA and TDLDA dynamical structure fators, and this prevents, therefore, plasmon contributionthe stopping power from being sensitive to the details ofwave-vector and frequency dependence of the responselarge velocities, well above the stopping maximum, all cculations presented in Fig. 1 converge to the well-knoBethe formula,34 in which the mean excitation potentialreplaced by the plasma energy.42

B. Position-dependent stopping power

We have carried out, from Eq.~3.16!, calculations of theelectronic energy loss versus impact parameter, and weobtained, within the RPA, the results presented in Fig. 3this figure we represent by solid curves the positiodependent stopping power of Al crystal for protons (Z151) moving withv50.2 a.u. along the~100! @Fig. 3~a!# and~111! @Fig. 3~b!# directions. In Fig. 3~a! the impact parametehas been taken to beb5(0,b,0)ac ~ac is the lattice constant!,with b from 0 to 1, and in Fig. 3~b! it has been taken to bb5(2110)ac . Contour density plots of the square latticcontaining both the projectile trajectory and theb vector aredisplayed in Figs. 4~a! and 4~b! for the ~100! and ~111! di-rections, respectively. The averaged electronic densalong the projectile paths depicted in Fig. 4 are plottedFig. 5, as a function of the impact parameter, showingthis fcc crystal the expected periodicity withb51/2, alsopresent in the position-dependent stopping power of Fig.

First of all, we note that the existence of small electrdensity variations in real aluminum results, through the odiagonal elements of the interacting density-response fution xG,G8(q,v), in non-negligible differences betweeposition-dependent and random stopping powers. We hobtained differences up to 20% for projectiles incident in

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~100! direction@see Fig. 3~a!#, and up to 10% for projectilesmoving in the~111! direction@see Fig. 3~b!#. The maxima inthe stopping power for trajectories along the interstitialgions (b51/4) and the minima near the cores (b50) areassociated with corresponding maxima and minima inintegrated electronic densities of Fig. 5.

A comparison with a LDA~Ref. 43! is also shown in Fig.3 ~dashed curves!. In this approach, the position-dependestopping power is obtained as the stopping power of amogeneous electron gas with an electron density equal toaverage electron density along the projectile path. Althouthe impact parameter dependence, as obtained in thisproach, is qualitatively the same as found within our fband-structure calculation, position-dependent correctionthe random stopping power appear to be largely underemated within the LDA. This is obviously a consequencethe stopping power depending not only on the local densbut also on the electron density gradients further away frthe projectile trajectory. We have also calculated positiodependent stopping powers of Al crystal for various nonrativistic velocities of the incident projectile, and we hafound that corrections to the random stopping power, as

FIG. 3. Position-dependent stopping power of Al crystal fprotons moving withv50.2, as obtained within the RPA. The projectile is incident in the (lnm) direction and the stopping power iplotted as a function of the magnitude of the impact vector alothe~abg! direction, with~a! ( lnm)5(100) and (abg)5(010); ~b!( lnm)5(111) and (abg)5(2110). Solid lines represent fulband-structure calculations, as obtained from Eq.~3.16!. Dashedcurves represent LDA calculations, obtained as explained intext, and the horizontal dashed line represents the stopping powa homogeneous electron gas withr s52.07. All results have beennormalized to the value of the random stopping power of Al crysfor the same velocity.

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PRB 58 10 313ELECTRONIC STOPPING POWER OF ALUMINUM CRYSTAL

tained within the LDA, are too small for all velocities. Thecorrections are found, within our full treatment of the bastructure, to slightly decrease with the velocity of the projetile. In the case of projectiles moving near the coresb50) along the~100! direction, we have obtained differencebetween position-dependent and random stopping poweapproximately 20%–12% in the velocity rangev50.2– 1 a.u. For projectiles moving near the cores (b50)along the~111! direction, these differences have been fouto be, in the same velocity range, approximately 10%–8

V. CONCLUSIONS

We have presented full band-structure calculationsboth random and position-dependent stopping powers oflence electrons in Al crystal, as obtained within linearesponse theory and with the use of the RPA andTDLDA. The random stopping power has been evaluated

FIG. 4. Contour valence-density plots of the square lattice ctaining the projectile trajectory (lnm) and the impact vector~abg!,with: ~a! ( lnm)5(100) and (abg)5(010); ~b! ( lnm)5(111) and(abg)5(2110). Projectile trajectories withb51/4 are also repre-sented.

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a wide range of projectile velocities. Our results indicate tat low velocities of the projectile band-structure effects resin the stopping power of real aluminum being about 7higher than that of a homogeneous electron gas with a dsity equal to the average electron density of the real cry(r s52.07). At velocities just above the threshold velocfor plasmon excitation, the stopping power of valence eltrons in Al crystal is found to be smaller than that of jelliuelectrons, corrections being of about 10%. In the higvelocity limit, the stopping power of both jellium and banelectrons is found to be the same and to coincide withwell-known Bethe formula.

The various contributions to the random stopping powof Al crystal have been calculated separately, showing tband-structure effects on the contribution to the stopppower coming from plasmon excitation are negligible. Thresult has been shown to be a consequence of the dynamstructure factor of both jellium and real aluminum fulfillinthe same sum rule, as long as the average electron densthe same. Also, there is, in the high-velocity limit, exaequipartition for the energy loss, as in the case of a homgeneous electron gas.

The position-dependent stopping power has been evated for projectiles with velocities up to the Fermi velociincident in various high-symmetry directions. We hafound differences between position-dependent and randstopping powers up to 10% for projectiles incident in t~100! direction and up to 20% for projectiles moving in th~111! direction. The magnitude of these position-dependrelative corrections to the random stopping power has bfound to be, within the LDA, largely underestimated, spcially at high velocities of the projectile.

ACKNOWLEDGMENTS

We thank P. M. Echenique for stimulating discussionWe also thank Alberto Garcı´a for his help in the calculation

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FIG. 5. Averaged valence densities along the projectile padepicted in Fig. 4, as a function of the impact parameterb. Thesolid line corresponds to a projectile trajectory along the~100! di-rection, as in Figs. 3~a! and 4~a!. The dashed line corresponds toprojectile trajectory along the~111! direction, as in Figs. 3~b! and4~b!. All results have been normalized to the value of the avervalence density of Al,n0 .

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10 314 PRB 58I. CAMPILLO, J. M. PITARKE, AND A. G. EGUILUZ

of the one-electron Bloch states. I.C. and J.M.P. acknoedge partial support by the Basque Unibertsitate eta IkerSaila and the Spanish Ministerio de Educacio´n y Cultura.A.G.E. acknowledges support from National Science Fo

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dation Grant No. DMR-9634502 and from the National Eergy Research Supercomputer Center. ORNL is manageLockheed Martin Energy Research Corp. for the U.S. DOunder Contract No. DE-AC05-96OR22464.

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38If more than 15G vectors had been taken in the calculation of E~3.14!, more than 60 bands would have been required for thestopping power with projectile velocities up tov54 a.u. to bewell converged.

39Contributions to the stopping power of a homogeneous elecgas are well known to come, within the RPA, from all wavvectors that are smaller than 2(qF1v), and our calculated re-sults of Fig. 1 are, therefore, expected to underestimate thestopping power of both jellium and band electrons for all veloties v.0.5 a.u. However, we have found that differences,large wave vectors, between the integrated dynamical strucfactor for jellium and band electrons are very small, andresults of Fig. 1 provide, therefore, an accurate measure ofeffect of the band structure on the random stopping powervalence electrons.

40J. Lindhard and A. Winther, K. Dan. Vidensk. Selsk. Mat. FyMedd.34 ~4!, 1–22~1964!.

41See, e.g., N-P. Wang and I. Nagy, Phys. Rev. A55, 2083~1997!,and references therein.

42The Bethe formula is, in the high velocity limit, nearly insensitivto differences among the plasma frequencies obtained invarious approximations to the response of the target.

43The use of this so-called LDA should not be confused withuse, within DFT, of the LDA for the evaluation ofVxc and thekernelKxc of Eqs.~2.5! and ~2.8!.