Charge states and energy loss of ions in solids

Nuclear Instruments and Methods in Physics Research B 93 (1994) 195-202 North-Holland

Brm Intemctious w&h blatarials & Atems

Charge states and energy loss of ions in solids A. Arnau Departamentode Firica de Materiales,Universidaddel Pais Vasco,Apartado 1072, San Sebastian20080, Spain Received 9 November 1993 and in revised form 30 December 1993

We discuss the mechanisms responsible for the energy loss of ions in solids to electronic excitations in two speed regimes: (i) low velocities (t’ < uF, ur being the Fermi vetocity), where the scattering of electrons at the Fermi level determines the stopping, and (iif intermediate velocities (around the stopping power maximum), where the relevance of the charge exchange of electrons between the target atoms and the projectile, as well as the dynamic screening of the interaction, is shown. 1. Intr~uction

A charged particle moving through matter loses a large part of its energy to electronic excitations, except in the very last part of its range where nuclear stopping is important. The electron excitation processes that contribute to the energy loss of the particle include charge exchange of electrons with the target atoms (capture and loss) as well as projectile and target electron excitations (including target ionization). The number of open channels is different for each particular projectiie-target combination and also depends on the relative velocity CU,) of the collision. Atomic units are used throughout this paper unless otherwise stated. At low velocities the most weakly bound electrons, of the target and projectile, are those which mainly participate in the energy loss process. When the projectile and target atomic numbers (2, and Z, respectively) are of similar magnitude, i.e. in a so called symmetric collision, the charge exchange of electrons is the dominant channel for the energy loss [I]. However, in an asymmetric collision CZ, +=K 2,) the outer shell electron excitation of the target electrons is the largest contribution to the energy loss of the moving ion. At high velocities the projectile acts as a point charge and it may be considered as a small perturbation to the state of the target electrons. The energy loss of the ion in this high speed regime depends smoothly on the target through the mean excitation energy (I) in a logarithmic term and a multiplicative factor containing the number of electrons (Z,) and the target atomic density (N), as was found by Bethe 121

(1) * Corresponding 212236.

author,

tel.+34

At i~terme~~te velocities, i.e. at and around the stopping power maximum, the problem is inherently much more complicated because almost all the channels are open and most of the processes cannot be treated accurately enough using perturbation theory. Another important point to be mentioned is the nature of the target; whether it is in the gas or solid phase and being an atomic or molecular species. These two points give rise to the so called phase and aggregation effects which is the topic of this workshop. As will be shown later for the case of a metal, the main difference between the solid and gas phases of the same element, as far as electronic stopping is concerned, appears at low velocities where the outer-shell (valence) electrons of the target and projectile participate actively in the processes that determine the energy loss of the penetrating ion. The reason for this difference can be summarized in one word: screening. In the gas phase the Coulomb interaction is long range while in the solid phase the screening mechanism reduces the range of the interaction, especially at low velocities. In the case of an insulator the screening by the valence eiectrons is not so efficient and we expect a weaker phase effect. However it is usually mixed up with an aggregation effect that appears for any molecular solid like the oxides SiO, and AlaO,.

2. Atomic versus solid state description

Let us concentrate now on the target excitations only assuming that the projectile is a point charge without any structure. This will be the actual situation in the weak coupling limit that we define as

43 216600, fax-t-34 43

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where un =ua(u,u,,) is the relative velocity of the collision, which is a function of the ion velocity u and the target electron velocity u,,. For the purposes of the discussion that follows it is enough to remember that ua = u when u >> L’,, and ua 2 u,, when u
3. Slow ions in solids The stopping power of an electron gas for a probe charge Z, may be written as [6] dE _=dx

22; rv2 I

dq q

qu /o

’

(3)

where Im( - l/e) denotes the imaginary part of the inverse dielectric function that depends on the momentum q and energy w transferred to the medium. This result is obtained using linear response theory; which

means that the screened potential is obtained just by linear screening of the bare Coulomb potential, i.e. V,,, = V~&E in Fourier space, and that the scattering amplitude is obtained in the first Born approximation, i.e. it is proportional to the Fourier transform of the potential. However, it is possible to improve on this result within the electron gas description by going beyond the first Born treatment of the scattering and performing a self-consistent calculation of the screened potential. In practice, a self-consistent calculation of the screened potential is only feasible at low projectile velocities (v < v,) where the spherical symmetry is still preserved. Let us concentrate now on the slow ion case. At low velocities the energy dissipation mechanism in the electron gas is the electron-hole pair creation at the Fermi level [7]. From the low frequency expansion of the inverse dielectric function Im(- 1,‘~) one can write the stopping power in terms of the Fourier transform of the static screened potential, 4PZ,

I/,,,(q) = ;. q*+?>

(4)

Before going into the details it is worth mentioning that 151 -1 Im aw when w-+0, (5) ( E 1 and so the stopping power is linearly proportional to the velocity at low velocities according to Eq. (3). If one interprets Eq. (4) in terms of non-relativistic scattering theory and makes the substitution Kscr((l) -+ -2rf(@>, (6) where f(e) is the scattering amplitude and q = 2v,sin(8/2) is the momentum transfer in the center of mass system for scattering angle 8, one obtains [8] dE = 2n,uv, I dx where dm = 2~ 1f(O) ( 2sin0 de. By introducing the transport cross section uttr(UF) = iTdg(e,

+)(I

- cos 0)

(8)

one arrives at the well-known result [9]: dE = n,uu,u,,( vF). dx The remaining part is the determination of the effective one-body scattering potential and the way of calculating the cross section. A full phase shift calculation is possible for spherically symmetric potentials using the partial wave expansion, and reads uttr(uF)

=

$ tco(Z+ 1) sin2[&(v,)- &+I(VF)I. (10)

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The first self-consistent calculation for the scattering potential using density functional theory in the local density approximation for exchange and correlation was performed by Echenique et al. [lo] for H and He ions in an electron gas. A thorough analysis of their proton data [11] for stopping powers and the experimental data shows that the agreement between theory and experiment is better than with previously obtained linear response theory results and that the velocity proportional stopping holds up to the Fermi velocity. Since then many calculations of low energy stopping power have been done using this method 112-151, including a generalization to interacting systems [16]. Briefly, one obtains the self-consistent potential from a set of one-body Schriidinger type equations [1’7]

3.5 2’

1 3.0 2.5

2.0

1.5 1.0 0.0

1.0

2.0

3.0

4.0

5.0

‘,

Fig. 1. Effective charge for a slow nitrogen ion in an electron gas as a function of the density parameter rs, See Eq. (13) in the text for the definition of the effective charge.

are shown in Fig. 2 and compared with the theoretical

in which the effective self-consistent tial depends on the density,

V,,[n] = - f

+ /dr’

scattering poten-

(12)

The exchange correlation part of the potential V,[n] contains all the static many-body effects. Brandt and coworkers [18,19] introduced the concept of an effective charge and were able to condense a great amount of stopping power data for ions with 2,~ 1. Echenique et al. 1121extended the earlier work for H and He stopping to ions with 2, > 2. They defined the effective charge in an operational manner as

(13) In Fig. 1 we show the effective charge of nitrogen (2, = ‘I) as a function of the density parameter rs defined from the valence electron density n, as

The effective charge varies from 2: = 7 for r, -+ 0 (high density limit) to Zf = 1.2 for r, = 5. Several experiments were performed 120-221 around twenty years ago to measure the energy loss of equal velocity slow ions (v < u,) of different atomic numbers Z, under channeling conditions. These data show pronounced Z, oscillations in the stopping power as a function of the ion nuclear charge, which were nicely reproduced using the self-insistent density functional method [12,16,23]. Results for the stopping power of best channeled ions along the (110) axial channel of Si

estimates. The agreement is quite good. Although silicon is not a free electron like material, it has a small band gap (e 1.1 eV) compared to the typical mean excitation energies of the electron-hole pair mechanism responsible for the energy loss at low velocities (uv, ff lo-15 eV for v a 0.5-0.8). These oscillations are related to the shell structure of the projectile ion’s

SicllO> 1

Q 0.8 0.6

0.4 0.2

Fig. 2. Friction coefficient (Q = S/u) for ions under channeling conditions along the (110) axial channel of Si as a function of the ion nuclear charge Z,. The open circles are experimental data from ref. (201,the closed circles, joined by a dashed line to guide the eye, are data from ref. [12]. These data are obtained with an average electron density for the channel corresponding to r, = 2.38 that is calculated by equating theory and experiment at Z, = 5. The solid line, from ref. [U], corresponds to a calculation using a local density approximation with a realistic density profile for the channel and no adjustable parameters.

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bound states, which is reflected for example in the minima around the closed shell values Z, = 2 and z, = 10. The stopping power for a slow antiproton may also be obtained with this method [15] and with a partially linearized Thomas-Fermi theory [24]. In Fig. 3 we plot the ratio of the proton to the antiproton stopping power at low velocities as a function of the parameter rs. At typical metallic densities (rs = 2) the ratio is roughly two. Although the experiments performed at CERN in the Low Energy Antiproton Ring (LEAR) to measure slow antiproton stopping power in solid targets have not been performed below 100 keV (around the stopping power maximum for most of the targets), it seems that the tendency of the measured data [25,26] is in agreement with these theoretical predictions. Furthermore, a one-parameter semiphenomenological model calculation by Nagy and Echenique [27] for antiproton stopping in an electron gas at arbitrary velocity shows very good agreement with the measured data. In Fig. 4 we plot the reduced stopping power (dE/dx)/(vZF) of an electron gas with r, = 3 for slow ions of different charges as obtained in density functional theory [28] and from a perturbative expansion in powers of Z, 129,301. The latter includes the higher order Z: (Barkas) term of the stopping power that is obtained using the second Born approximation for the scattering amplitude to describe the scattering of electrons by a Yukawa potential and agrees with the result of a many-body higher order perturbation theory calculation by Hu and Zaremba [31]. The results show that the perturbative description has a very limited range of validity. It is formally exact in the limit Zr/o, e 1. To be precise, a good criterion to estimate the validity of

rs 5 3

-0.050

-2.0 -1.5 -1.0 -0.5

0.0

0.5

1.0

1.5

2.0

Fig. 4. Reduced stopping power (Q/Z:) as a function of Z, for an electron gas with r,= 3. Curve (a) is the result of a second Born approximation for electron scattering on a Yukawa potential and curve (h) refers to a DFT calculation in which exchange and correlation effects have been neglected (Hartree-only). The DFT result with inclusion of exchange and correlation effects is shown as curve cc).

the perturbative writing 1 dE ; dx =AZ;

expansion

+ BZ:,

could be established

by

(15)

and it is that B/A < l/10. A similar perturbative expansion in the high velocity limit (Z,/U < 1) is also questionable for extrapolation down to the stoppingpower maximum. Sometimes it is better to keep just the leading term in the perturbation series rather than correcting it by higher order terms which have the same order of magnitude.

P

z-

4. Energy loss at intermediate

2.0

I.5

1.0 I

2

3

I I

,O.Y.,

Fig. 3. Ratio of the stopping powers of a homogeneous electron gas for slow protons and antiprotons as a function of rs.

velocities: charge states

In the previous section we have presented results for the stopping power of slow ions obtained within density functional theory as applied to the problem of a static impurity in a homogeneous electron gas. Thus, they are exact in the limit of vanishing projectile velocity (u within the approximation of substituting the valence electron excitations of the target by that of a homogeneous free electron gas. Consequently, any effect related to inhomogeneities of the electron density or the presence of the lattice ion cores cannot be taken into account in this model. This is the reason why the theoretical results are usually compared to data obtained under channeling conditions, where the electron density is nearly homogeneous and the interaction with the lattice ion cores is strongly reduced.

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At low ion velocities the charge state of the ion is determined by the strength of its nuclear charge (Z,) and the valence electron density (n,) of the host that reflects the screening by the metal electrons. These latter will be sometimes occupying bound states and others continuum states in the conduction band, but in an ideal metal there will be always a total screening, i.e, the induced charge exactly cancels the impurity charge [32]. This result is expressed by the Friedel sum rule f33] for the phase shifts at the Fermi level:

(16) As the ion velocity increases the charge state of a positive ion will be no longer determined by Z, and n, only, but, as long as the probabili~ of loosing bound electrons is not negligible, a competition between the processes of capture and loss of electrons by the ion will determine the charge state distribution and this will be obviously a function of the ion velocity (u). A self consistent calculation within density functional theory is not feasible; the breakdown of spherical symmetry and the time dependence of the response overcomplicate the calculation. A hydrodynamical approach has been used by Arnau and Zaremba [34] to study nonlinear screening and stopping of positive and negative charges in an electron gas at finite projectile velocities, while Nagy et al. [35] have used an extended comas-Fe~i theory. Also the effects due to the interaction with the lattice ion cores, like inner shell capture and ionization or electron loss by the moving ion due to the interaction with the lattice, start to play a relevant role as the ion velocity increases. At very high velocities screening by the valence electrons and the crystal structure are not important in determining the charge states. One may simply use the gas phase capture and loss cross sections to determine the charge states of the ions in solids [361. The problem is that the knowledge of the charge state distribution is not enough to calculate the stoppjng power. One also needs to evaluate the. energy loss in charge exchange processes and the partial stopping powers of each charge state. In the charge-state approach [3?] the stopping power for a charge state equilibrated beam of ions is given by 1381

where I#J~is the probability of finding the ion in charge state i, Si is the partial stopping power for charge state i, Ct&cY,PII and wij(cr,b) are the cross section and the co~esponding transition energy for the electronic process in which the projectile charge state changes from i to j while the electrons of both the target and projec-

199

tile evolve from state a to p. The latter should include, in principle, target excitation and ionization with projectile capture, loss and excitation. In practice, one has to restrict the number of open channels in each particular case to arrive at a tractable calculation. In the case of light ions (Z, Q 2) on heavy targets (Z,a 10) the sum of the partial stopping powers is the major ~ntribution to the totai stopping power (typically larger than 70%). I-Iowever, in the case of symmetric collisions (2, s Z,) or heavy ion projectiles and targets (Z,,Z, > 10) the charge exchange of electrons is an important channel for the energy loss [39,40]. The calculation of the partial stopping powers (Si) of the different ion charge states may be done using different appro~mations depending on the velocity range where the charge state fraction #* is appreciable (low or high velocity limits). Finally, the energy loss in projectile inelastic processes and simultaneous target excitation or ionization is difficult to evaluate due to the lack of knowledge of accurate values for the relevant cross sections. For H and He ions interacting with solid metal targets we consider the following charge exchange mechanisms [4Ij: (i) Inner shell electron capture. Electrons from deep levels of the target atoms may be captured by fast ions in atomic type processes. The cross sections for these processes do not depend very much on solid state effects like screening, due to the fact that small impact parameters dominate the relevant transition amplitudes. Atomic collision theory techniques are then used to calculate these cross sections. One needs an accurate description of the initial and final state electron wave function to calculate the transition amplitudes. A proper choice of a weak perturbation allows one to use a first order approach Iike the continuum distorted wave (CDW) method [42]. Capture processes may be seen as a particular case of target ionization by the projectile in which the final state has a large probability of being around the projectile, i.e. it is essentially a projectile bound state. (ii) Electron loss by the projectile. The interaction with the lattice ion cores and with the electrons at the Fermi sea produces electron stripping of the projectile ion. (iii) Electron capture in three body recombination processes (TBRP>. An electron at the Fermi sea may be captured by the moving ion assisted by a third body (creation of a plasmon or an electron-hole pair) due to the electron-electron interaction [43] which is screened. These processes taking place at zero velocity are the well known Auger capture processes to the conduction band electrons. Amau et ai. 1441and Perialba et al. [445]have calculated the stopping power of He and II ions in aluminum. Screening by the valence electrons prevents

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in Hrys. Rex B 93 f‘I994) 195-202

Fig. 5. Stoping power (in a.u.) as a function of ion velocity for He ions in aluminum. Curve (a) is the He0 contribution, curve (b) is that of the He+ charge state and curve (d) corresponds to the He ‘+ charge fraction. Curve (c) is the contribution from capture and loss processes, The thick solid line is the sum of all the others.

the existence of more than one bound state (1s type orbital) around the projectile in this case. The wave function and binding energy of the state may be obtained within the self-energy approach (431 of Guinea et al. that explicitly includes the dynamic screening of the interaction, by using a variational procedure to minimize the energy of the ion plus bound electron composite in the electron gas system. Within this method it is also possible to obtain the cross sections for capture in TBRP and loss due to the interaction with the electron gas, i.e. the inverse process of TBRP capture. Capture of inner shell electrons is obtained using atomic collision theory methods [46-491, while electron loss cross sections in the processes of interac-

10

p

10

100

E (keV/u)

loo0

Fig. 6. Stopping cross section (lo-*’ eVcm’) as a function of projectile energy (keV/u) for H projectiles in Zn targets. Curves (a) and (b) are the best fit to rhe measured data for the gas and solid phase, respectively. Curves cc.1and (d) are the result of a calculation using the charge state approach for the gas and solid phase, respectively.

tion with the lattice ione cores are calculated in first order perturbation theory with a proper choice for the screened interaction pseudopotential [SO].The capture and loss cross sections are used both to calculate the charge state distributions and the energy loss in charge exchange processes. The partial stopping power of each of the charge states (Sj) are calculated as follows. At low velocities (low charge states) we use the nonlinear density functional approach [51] to calculate the stopping power of the corresponding charge state from the transport cross section at the Fermi level. At high and intermediate velocities we use first order approaches for a point charge or a dressed projectile [521 (e.g. He+) in the case of He ions, or evaluate the transport cross section at different relative velocities for scattering of electrons by the screened potentials [45] in the case of H ions. In Fig. 5 we plot the stopping power of aIuminum for He ions at different velocities. The different contributions to the total stopping power are shown separately. At low velocities the He0 charge state dominates, while at high velocities the He’+ charge state is the major contribution to the energy loss. The charge exchange processes amount up to a 15% of the total energy loss around v = 2u,. The stopping cross section for H projectiles in gas and solid Zn targets has been measured and explained using the charge state approach, by Bauer et al. [53]. The effect of the solid target 3d electrons is taken into account in an approximate manner by considering them to be in localized atomic orbitals, but the energy level is shifted according to band structure calculations 1541. The target and projectile contributions to the energy loss are calculated for both phases, and the stopping cross section for the gas phase is found to be larger as compared to the solid, in agreement with experiment. These results are shown in Fig. 6. The 4s and 3d target

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electron excitations are the dominant channel for the energy loss of H ions in both the two phases of Zn in the energy range 20 d E sz 700 keV. However, the screening of the Coulomb interaction by the 4s conduction band electrons in the solid phase reduces so much its strength that it is responsible for the difference at low energies (E z 50 keV), where the phase effect is more pronounced.

Acknowledgements We gratefully acknowledge financial support by Euskal Herriko Unibertsitatea, Gipuzkoako Foru Aldundia, Eusko Jaurlaritza and the Spanish DGICYT. Fru~tfui and stimulating discussions with P.M. Echenique, I. Nagy, A. Salin, P. Bauer, V.H. Ponce and G. Schiwietz are also acknowledged.

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