Electronic Stopping of Slow Protons in Transition and Rare Earth Metals: Breakdown of the Free Electron Gas Concept

The electronic stopping cross sections (SCS) of Ta and Gd for slow protons have been investigated experimentally. The data are compared to the results for Pt and Au to learn how electronic stopping in transition and rare earth metals correlates with features of the electronic band structures. The extraordinarily high SCS observed for protons in Ta and Gd cannot be understood in terms of a free electron gas model, but are related to the high densities of both occupied and unoccupied electronic states in these metals.

When ions propagate in matter, they are decelerated due to interaction with both, nuclei and electronsin other words, by nuclear and electronic stopping, respectively. The mean energy loss per path length is given by the deceleration force, i.e., the stopping power S = dE/dx. To describe the interaction with atoms or molecules, often the stopping cross section (SCS)  = 1/n·dE/dx is used, where n represents the atomic or molecular density of the target material, respectively. Precise knowledge of electronic stopping is important in many different fields like outer space exploration (space weathering), nanotechnology (ion beam patterning), fusion research (plasma-wall interactions), or medicine (radiation therapy) [1,2,3,4]. Thus, the SCS is a quite fundamental quantity, for which it is mandatory to have a detailed understanding of the prevailing energy loss mechanisms.
At high ion velocities v >> vF (vF denotes the Fermi velocity of the target electrons), energy loss of ions is dominated by electronic stopping. In this regime, light ions represent only a weak perturbation for the target electrons, and first order theoretical models are sufficient for a precise description of the energy loss process [5,6,7,8]. For low ion velocities v  vF, both electronic and nuclear collisions contribute to the stopping power. In this regime, the electronic energy loss is predominantly due to interaction with valence electrons. For a free electron gas (FEG), the stopping power is proportional to the ion velocity, S = Q(Z1,rs)·v, where the friction coefficient Q depends on the atomic number of the ion Z1 and the Wigner-Seitz radius of the FEG rs = (3/4πne) 1/3 , with the FEG density ne = Nval·n, where Nval stands for the (effective) number of valence electrons per atom [9]. At low velocities, even for light ions, the electronic interaction is so strong that nonlinear theoretical models have to be applied to solve this complex many-body problem. In their seminal work, Echenique et al. found the quantum mechanically correct solution for the stopping power of static ions in a FEG [10].
When Mann et al. showed that for many target materials the existing experimental data could be well reproduced for ion velocities up to vF by this nonlinear model, the physics of slow ion stopping was seemingly well understood [11]. 4 Limitations of the applicability of the FEG model have been discovered for noble metals at very low proton velocities v << vF: deviations from velocity proportionality of , e.g., for H + in Au at v  0.2 a.u. [12] were attributed to the finite excitation threshold of the d band (for Au, 2 eV [13,14]). Recently, time-dependent density functional (TD-DFT) calculations of electronic stopping of protons in Au in channelling geometry confirmed this interpretation and revealed a low but finite contribution of the d band to electronic stopping down to v = 0.15 a.u. [15]. Still today, even for protons traversing in a real metal (beyond a FEG) it is an open question which are the processes dominant in the stopping power at low velocities [16].
In fact, for noble metals, there exist two distinct regimes where the non-linear FEG model [10] can be applied by assuming a proper FEG density. For Au, at proton velocities v < 0.2 a.u., electronic stopping is predominantly due to excitation of the 6s 1 electron, which can be described as a low density FEG (rs = 3.01 a.u.), whereas at v ≈ vF, all conduction electrons are excited as a FEG of effective density as deduced from the plasmon frequency (rs,eff = 1.49 a.u.) [17,18]. Metals like Pt feature a high density of states (DOS) up to the Fermi energy EF due to d electrons, followed by a low density of free states at E > EF. For Pt, the use of rs,eff = 1.63 a.u. (Nval,eff = 5.6) leads to good agreement between experimental and FEG stopping power at all velocities up to vF [17,19].
In this Letter, we present evidence that the FEG model fails when applied to metals with d or f bands of high DOS below and above EF, like in transition and rare earth metals. To this aim, we present electronic SCS data for low-velocity protons in polycrystalline Ta ([Xe]4f 14 5d 3 6s 2 ) and Gd ([Xe]4f 7 5d 1 6s 2 ); for Gd, the velocity range was extended to cover also the stopping maximum.
Since transition and, especially, rare earth metals are chemically reactive, information on impurities and their chemical states is essential. To this aim, purity of both samples (250 µm 99.99 % foils of Ta and Gd, respectively) was analyzed, by time-of-flight elastic recoil 5 detection (TOF-ERD) and Rutherford backscattering spectrometry (RBS) at Uppsala University, as well as by x-ray photoelectron spectroscopy (XPS) and Auger electron spectroscopy (AES) in ZONA (JKU). For Ta, XPS and TOF-ERD showed no significant bulk concentrations of light impurities. Also the Gd foil was found to be a pure metal covered by a ~ 1 µm thick nearly stoichiometric oxide layer. After removing the Gd2O3 layer, very fast oxygen uptake was observed even at UHV conditions (base pressure: 5  10 -10 mbar), in accordance with [20,21]. XPS indicated that also with adsorbed oxygen on top Gd is present in metallic form. The energy loss measurements were performed at the IEP in Linz employing the UHV time-of-flight low energy ion scattering (TOF-LEIS) setup ACOLISSA [22] and the RBS setup of the AN 700 accelerator [23]. In ACOLISSA, both samples were cleaned by means of 3 keV Ar + sputtering in order to remove adsorbed hydrocarbons and the native oxide TOF-LEIS spectra of Gd, Ta, and Au were acquired by use of beams of hydrogen and deuterium ions (monomers and dimers) in the range of 0.5 keV/u -10 keV/u (scattering angle θ = 129°) at a base pressure of < 4  10 -11 mbar. Even for Gd, no change in the LEIS spectra was observed when comparing the first and the last measurement within one data collection cycle. To cover the Bragg peak of Gd, RBS spectra of Gd and Au were recorded employing atomic beams of H + and D + in the range 30 keV/u -600 keV/u (θ = 154.6°).
The SCS of the materials of interest ("X") was deduced from the energy spectra for projectiles backscattered from X and from Au as a reference sample ("ref"). This approach makes use of the fact that the ratio of the spectrum heights, NX/Nref, contains information on []ref/[]X [24].
Because of the influence of multiple scattering at low energies, the ratios of the experimental 6 spectra in an energy interval close to the kinematic onset [25] were compared to the results from corresponding Monte Carlo simulations (TRBS, [26]). In the simulations, the ZBL potential [27] was used, and Au was set to match the experimental value (SCS data from [14]), leaving X as the only parameter, which was optimized until convergence was reached; i.e., the heights of the resulting spectra changed by less than 3 %.
Since RBS spectra of the polished Gd sample showed an oxygen enriched surface layer of considerable thickness [28], the LEIS data were evaluated applying Bragg's rule [29] taking the SCS data of oxygen from SRIM-2013 [30]. This correction results in a lowering of Gd by 8 %. In case of the RBS data, energy windows in the evaluation were chosen such that only projectiles backscattered in the bulk Gd were considered. For E > 200 keV, spectra were simulated by use of the program SIMNRA [31] employing a screened Coulomb potential [32] and the dual scattering model.  Fig. 2. LEIS and RBS data are very well consistent. No isotope or vicinage effects were observed; thus, we refer in the following discussion for the projectiles as to protons. For Gd, the maximum SCS is found to be very high (Gd ~ 48  10 -15 eV cm 2 /atom), while the position of the stopping maximum is rather low (Emax ~ 80 keV). At higher energies, the RBS data coincide with data from literature [33,34,35,36]. The bold solid line is a fit using the semiempirical model described in [27]. The overall consistency of the results and the agreement with literature gives confidence in the experiment and the evaluation procedure, even for a chemically reactive metal like Gd.

FIG. 2 Electronic SCS of Gd for H ions (protons and deuterons) are shown as function of the proton energy in the range 0.5 keV -10 MeV (LEIS data: full circles, RBS data: full triangles). Also displayed is a fit to the complete data set. Open symbols refer to data from
Refs. [33 -36].
In Fig. 3, the low-velocity SCS of Gd, Ta, Pt, and Au for protons are presented as function of the ion velocity. The SCS of both, Gd and Ta are considerably higher than that of Pt and Au: at v = 0.2 a.u., Gd and Ta exceed Au by a factor of 3.8 and 2.6, respectively. The statistical uncertainty of the SCS in the LEIS regime is estimated to be < 10 % per data point.
Systematic errors are estimated to be < 10 % and mainly due to the correction for impurities.
The statistical uncertainty of the present RBS data is 5 % per data point; systematic errors due to the correction for impurities are < 5 %.

FIG. 3 Electronic SCS of Au, Pt, Ta, and Gd for protons are displayed as function of the ion velocity in atomic units. The upper abscissa denotes the corresponding proton energies.
While at v < 0.2 a.u., the nonlinear FEG model [10] successfully describes the stopping ratio Pt/Au  1.7 by use of effective FEG densities [17], for Gd and Ta, the FEG concept fails: by use of QH (rs) from theory [10,37], the slopes of the experimental data can be converted to For Gd, the value for rs,expt is almost identical to that of Al [40]. Hence, SGd  SAl, in accordance with the fact that Nval = 3 for both Al and Gd. However, the atomic density nGd is lower than nAl by a factor of 2. Consequently, rs,expt = 2.03 corresponds to Nval,expt = 6.4 for Gd in clear contradiction to the FEG model, on condition that the seven f electrons do not participate in the electronic interactions of slow protons (neither in the e-h pair excitation nor in the plasmon excitation). Note that the experimental plasmon energy of Gd (ħp,Gd = 14 eV [41]) corresponds to rs,eff = 2.24 (Nval,eff = 4.7).
Since the observed features in electronic stopping should be related to electronic properties of the materials, the electronic DOS of the investigated metals were calculated by means of DFT, employing the VASP code [42,43]. For Au, Pt, and Ta, the PBE exchange-correlation functional has been used [44]. In the case of Gd, a PBE + U approach was used with the recommended values U = 6.7 eV and J = 0.7 eV [45,46] in order to assure an adequate description of the 4f orbitals. In all cases, the core-valence electrons interaction is treated by the projected augmented wave (PAW) potentials [47,48]. The energy cutoffs for the plane wave basis sets are 287.4 eV for Au, 287.9 eV for Pt, 279.6 eV for Ta, and 320.6 eV for Gd.
The Brillouin zone is sampled by a 17  17  17 Monkhorst-Pack grid of k points [49] for Au, Ta, and Pt, and a 17  17  11 grid for Gd. In evaluating the DOS, the occupancies of the electronic states are determined with the tetrahedron method. In Fig. 4, the electronic DOS of Au, Pt, Ta, and Gd are displayed as a function of E -EF.
Amongst these metals, only for Au, the DOS at E  EF is dominated by the s electrons, while for the other metals, the density at E < EF is considerably higher, due to the contribution of the d bands. In our calculation, the occupied f states of Gd are located at  8.3 eV below EF. From this, one may assume that at v  0.2 a.u., the f electrons do not participate in electronic stopping, similarly as the d band in Zn is not excited at these low velocities [17]. On the other hand, an increase of the slope of Gd is not observed at v > 0.2 a.u. (as for Zn). Thus, excitation of the f band of Gd is either possible already at very low velocities or becomes considerable at higher velocities only. The conduction band of Gd has parabolic shape similarly as for FEG like metals, e.g., Al, but the DOS (EF) of Gd exceeds that of Al by a factor of 4.5, while the occupied part of the band is much narrower. At  3 eV above EF, the DOS exhibits a prominent peak corresponding to seven electrons due to the f band, which may increase the efficiency of the e-h pair excitation at very low ion velocities (e.g., v ≈ 0.1 a.u.) even further. This may cause a steeper slope of the Gd SCS data at v < 0.25 a.u..
As a next step, we compare the DOS of occupied and unoccupied states of the metals presented in Fig. 4. The DOS integrated from -10 eV up to EF yield values in the range from 5 electrons (Ta) to 11 electrons (Au), as expected from Nval. Above the Fermi level, the DOS is low for Pt and Au (s-and p-states only), while Ta and Gd exhibit a high d-electron density; the DOS integrated from EF up to 10 eV ranges from 3.3 electrons (Au, Pt) and 6.9 electrons for Ta to  17 electrons (Gd). Note that for Ta, the DOS has nothing in common with a FEG like parabola, neither below nor above EF. Given these facts, our stopping data can be qualitatively understood in the following way: on the one hand, Gd and Ta exhibit high DOS below and above EF, thereby allowing for low energy excitations with high probabilities. On the other hand, for Pt and Au, the low DOS above EF will be responsible for the low SCS, as compared to Ta or Gd. Note that this argument should hold true for the other transition and rare earth metals too, as long as both, DOS (E < EF) and DOS (E > EF) are high. In fact, also at the stopping maximum, the combination of high DOS below and above EF helps to explain the very high SCS of Gd (excitation of f electrons).
To conclude, electronic stopping of slow protons in transition and rare earth metals is so efficient that any FEG model fails to describe the slowing down process, ending up in absurdly high effective FEG densities, or in other words, in an unphysical number of valence electrons. For instance, both the rare earth metal Gd and aluminum exhibit Nval = 3 and virtually the same FEG density parameter (rs,expt  2.03) deduced from the measured SCS, but 13 for Gd, an effective number of 6.4 valence electrons results due to its low atomic density. It is interesting to note that the DOS at EF is much higher for Gd and Ta compared to Au and Pt.
For the latter, the low DOS above EF may be responsible for the much lower values of Nval,eff (for Pt, 5.6 instead of 10). Therefore, a FEG description of electronic stopping collapses when applied to transition and rare earth metals featuring a high DOS (below and above EF).
Elaborate many-body theoretical models like TD-DFT are required to thoroughly understand the underlying physical mechanisms and to find possibly a way of predicting realistic stopping power values for rare earth and transition metals even for simple ions.