Core electrons in the electronic stopping of heavy ions
Electronic stopping power in the range is accurately calculated from first principles. The energy loss to electrons in self-irradiated nickel, a paradigmatic transition metal, using real-time time-dependent density functional theory is studied. Different core states are explicitly included in the simulations to understand their involvement in the dissipation mechanism. The experimental data are well reproduced in the projectile velocity range of . The core electrons of the projectile are found to open additional dissipation channels as the projectile velocity increases. Almost all of the energy loss is accounted for, even for high projectile velocities, when core electrons as deep as are explicitly treated. In addition to their expected excitation at high velocities, a flapping dynamical response of the core electrons is observed at intermediate projectile speeds.
The dissipative processes in ion irradiation of matter are of primary interest from the basic physics point of view (a paradigmatic example of strongly non-equilibrium but quasistationary processes) as well as for technological applications (aerospace electronics Bagatin and Gerardin (2015), future energy application materials Granberg et al. (2016), radiation based cancer therapies Levin et al. (2005), and material science Townsend (1987)). An exact description of these processes means solving a full time-dependent quantum many body problem involving electrons and nuclei, which is a computationally prohibitive task. A multitude of approximate models have been developed and applied to this problem since the early years of quantum physics. Following the pioneering work of Fermi and Teller Fermi and Teller (1947), Lindhard in 1954 Lindhard (1954) and Ritchie in 1959 Ritchie (1959) applied a linear response formalism to study the energy loss in simple metals. In 1976, Almbladh et al. Almbladh et al. (1976), and from 1981 onward Echenique et al. Echenique et al. (1981, 1986, 1990) have used density functional theory (DFT) to calculate the nonlinear response of the electron gas to the perturbation produced by a swift ion. Ever since, the problem has been extensively studied using various approaches Campillo et al. (1998); Pitarke and Campillo (2000); Juaristi et al. (2000); Nagy and Aldazabal (2009); Winter et al. (2003); Martin-Gondre et al. (2013); Koval et al. (2013); Sigmund (2014). The success of the mostly analytical models remained limited in their practical applicability to simple metals and simple ions Roth et al. (2017a, b). However, a considerable progress has been made thanks to the advances in electronic structure methods Hohenberg and Kohn (1964); Runge and Gross (1984) and the availability of high throughput computational resources. Recently, electronic dissipation under ion irradiation in different realistic systems has been studied within a time-dependent tight-binding method Mason et al. (2007); Race et al. (2010, 2013), linear-response time dependent density functional theory (LR-TDDFT) Shukri et al. (2016) and real-time (RT)-TDDFT Pruneda et al. (2007); Krasheninnikov et al. (2007); Quijada et al. (2007); Hatcher et al. (2008); Correa et al. (2012); Zeb et al. (2012); Ojanperä et al. (2014); Ullah et al. (2015); Schleife et al. (2015); Wang et al. (2015); Quashie et al. (2016); Lim et al. (2016); Reeves et al. (2016); Li et al. (2017); Yost et al. (2017); Bi et al. (2017).
The pioneering work presented in Refs. Pruneda et al. (2007); Hatcher et al. (2008); Correa et al. (2012); Zeb et al. (2012) is not only in good agreement with experiments and provides further insights into the problem of electronic stopping of ions, but also demonstrably sets the stage for using RT-TDDFT to study this problem in a wide variety of systems. However, most of the previous RT-TDDFT based studies have been limited to simple projectiles (H, He) Yost et al. (2017) and low projectile energies, giving low electronic stopping values. In most of these cases, the electronic stopping power, which is the energy lost by the projectile per unit path length, , is of the order of and at such energy deposition rates very little or no permanent damage is expected. The effect of explicit treatment of the core and semi-core electrons of the target with light projectiles on has been studied using LR-TDDFT Shukri et al. (2016) and RT-TDDFT Zeb et al. (2012); Schleife et al. (2015). Ojenperä et al. Ojanperä et al. (2014) have shown the significant effect of core electrons of the projectile using RT-TDDFT.
The self-irradiated transition metals are known to have much higher values of Ziegler et al. (2010), in the range of , which can cause significant permanent damage, mainly in the form of ion-tracks de la Rubia et al. (1987). These systems have not been studied using first-principles methods before, including the full effect of core states. In fact, no material with values in the range has ever before been simulated beyond linear response. The physics of these systems remains poorly explained and quite challenging to study within first principle approaches. The excitation of core states and their contribution in dissipation is expected to be critical in explaining extremely high stopping powers Caro et al. (2017). The precise mechanism of these excitations and their relative contributions remains poorly understood.
We have considered the prototypical problem of a self-irradiated transition metal, nickel, in which a primary knock-on atom (PKA) shoots through the bulk. This is a common occurrence in materials exposed to neutron radiation. Ni based alloys are known for their radiation tolerance Lu et al. (2016), thermal stability and optimal mechanical properties, making them promising candidate materials for next generation energy and aerospace applications Jin et al. (2016); Zhang et al. (2016); Levo et al. (2017). The presence of Ni in structural alloys is known to play an important role in mitigation of swelling under irradiation Bates and Powell (1981). Nickel, along with iron and tungsten, is the subject of extensive radiation damage research Osetsky et al. (2015); Granberg et al. (2016); Zhang et al. (2017). Most of the radiation damage studies are limited to classical and adiabatic molecular dynamics simulations, but an accurate description of radiation damage demands a good characterization of non-adiabatic electronic contributions. They become very pronounced in the case of heavy projectile at relatively higher velocities.
There are no direct experimental data available for the stopping power of Ni in Ni, except for the element-wise interpolations of Stopping and Range of Ions in Matter (SRIM) model Ziegler et al. (2010) which makes the prediction of our simulations ever more important. The SRIM model shows that in self-irradiated Ni, nuclear stopping is dominant for velocities up to 1 atomic unit (a.u. hereafter) and quickly diminishes beyond it (see Fig. 2). However, becomes dominant above of velocity and accounts for almost all of the stopping power in the high velocity regime. In this work we have considered the velocity range from to which includes the maximum of electronic stopping.
TDDFT is a reformulation of the many-body time-dependent Schrödinger equation Runge and Gross (1984) analogous to what DFT is to the time-independent Schrödinger equation Hohenberg and Kohn (1964). Using the Kohn-Sham scheme, the many-body time-dependent problem is effectively reduced to a one-body problem in an effective potential Kohn and Sham (1965). TDDFT is, in principle, exact; but in practice the exchange and correlation part of the effective potential is approximated using different schemes. In RT-TDDFT, the one-body Kohn-Sham wavefunctions are explicitly propagated in time, unlike what happens in the linear response approaches, which work in the frequency domain. We have used the RT-TDDFT formalism within the adiabatic local density approximation (ALDA) Ceperley and Alder (1980) for exchange and correlation, using the first-principles code qb@ll Gygi (2008); Draeger et al. (2017) for our calculations, as described in Ref. Schleife et al. (2012).
The Kohn-Sham wavefunctions represent individual electrons and are expanded in a plane-wave basis. changes less than 3% in the worst case as the energy cutoff is varied from to . An energy cutoff of is used for the rest of the calculations. The ions are represented by norm-conserving non-local pseudopotentials, factorised in the Kleinman-Bylander form Kleinman and Bylander (1982). A supercell containing atoms was constructed by conventional cubic cells of Ni. The experimental value of for the lattice constant was used.
The simulations could be described as virtual experiments. A Ni interstitial is placed inside the supercell and a self-consistent ground state is obtained. The self-consistent ground state serves as an initial state for the real-time evolution of the Kohn-Sham wavefunctions. From the self-consistent ground state, the Ni interstitial is instataneously given a velocity at mimicking a PKA event, hence becoming a projectile. As the projectile shoots through the bulk, the Kohn-Sham wavefunctions are propagated in time using a fourth order Runge-Kutta integrator Schleife et al. (2012), with all atoms fixed except the projectile, which moves with a constant velocity. The constrained ionic motion is based on the fact that ionic velocities, for the considered simulation times and trajectories, do not change significantly. After testing the convergence of simulation parameters, a time step of or smaller is used for time-integration ( by additionally requiring ). The sudden kick causes a relatively short-lived transient before the system enters a dynamical steady state. The total Kohn-Sham energy of the electronic sub-system is recorded as a function of distance travelled by the projectile. The constrained motion of ions guarantees that the change in the electronic total energy along the trajectory corresponds to the ‘electron-only’ stopping () experienced by the projectile. The Kohn-Sham energy as a function of distance is recorded for different velocities. The slope of each of those curves is obtained by simple linear curve fitting as detailed in Refs. Ullah et al. (2015); Schleife et al. (2015); Quashie et al. (2016), which gives for that particular velocity. The calculations in this work are in channeling condition along the  direction of the face-centred cubic crystal of Ni.
We have investigated the contribution of core-states by controlling their inclusion via a sequence of different pseudopotential approximations. The pseudopotential approximation replaces core electrons by an effective potential that defines the physics of the valence electrons. It is, in general, a necessary approximation when working with a plane wave basis Kohanoff (2006). The core states frozen into a pseudopotential cannot polarize or take part in any dynamic process. Redefining the partition between valence and core electrons allows us to assess the pseudopotential approximation. We have exploited this freedom to study the participation of the different core states in the process of energy deposition. We have generated four pseudopotentials, namely, Ni10, Ni16, Ni18, and Ni26 with different valence electrons, as defined in Table 1 Fuchs and Scheffler (1999); Rappe et al. (1990); Grinberg et al. (2000).
The results of our calculations, for the different core/valence sets, are presented and compared with the SRIM data in Fig. 2. The calculated of Ni10 in Ni10 (Ni projectile and host atoms all with 10 explicit electrons) is clearly underestimated in practically the whole velocity range investigated, (open triangles), by about an order of magnitude as compared to SRIM data (solid line). Not only the is underestimated, the maximum of occurs around of velocity while SRIM predicts it to peak around . However, redefining more electrons from frozen core to explicitly simulated valence states makes a very significant difference. In a similar calculation with a Ni16 projectile in Ni16 bulk, the calculated increases almost by a factor of two, as shown by the solid squares. This is a strong direct evidence of the importance of core states in the energy dissipation mechanism. However, the remains underestimated in comparison to the SRIM data. Digging further in the same direction; we have calculated the of Ni18 in Ni18 bulk and Ni26 and Ni26 bulk. The Ni18 projectile in Ni18 bulk calculation (solid circles), confirms the trend, although it does not fully account for the underestimation in the . The Ni26 projectile in Ni26 bulk case (open squares) produces the , in perfect agreement with the SRIM data from to of velocity, while it is underestimated by less than 10% between to , which is within the anticipated inaccuracy in the SRIM model for heavier elements Ziegler (2004).
In addition to the good agreement with the SRIM model based data, these results provide a very clear evidence that core states as deep as very significantly affect the of the swift ions. The values for different valence states converges in the low-velocity limit, but those for limited valence states saturate too early. The smaller the number of valence electrons, the earlier the saturates with increasing velocity. Very importantly, Fig. 2 also reveals that if the right number of core electrons are allowed to participate in the dynamic processes, almost all of the dissipation can be accounted for within the RT-TDDFT formalism.
To distinguish the effect of core electrons in the host from those of the projectile, we have computed the of a Ni26 projectile in a Ni18 host (open squares). It is very interesting to note that it almost exactly matches the of the Ni26 in Ni26 case. The only difference between Ni18 in Ni18 (solid circles) and Ni26 in Ni18 is the presence and consideration of as dynamical electrons of the projectile, which increases the by a factor of almost two pointing to the importance of bare charge of the highly ionized projectile. This result strongly suggests that the critical contribution comes from the electrons of the projectile while the deep electrons of the host do not make a significant difference.
Regarding the position of the peak of the curve, as more core electrons are treated explicitly, the peak position gradually corrects by shifting rightwards. The SRIM data predicts the peak position around of velocity, while our calculations put it around of velocity, a 15% smaller value.
The case of Ni26 projectile in a Ni18 host allows us to characterize the dynamics of the core electrons of the projectile. In Fig. 3 we show the time evolution of the energy expectation values of the occupied Kohn-Sham orbitals for different projectile velocities. We show different electronic levels and bands, and the three lowest corresponding to the initially occupied levels of the projectile (Ni26). Although the calculation of is well converged with respect to the energy cutoff, the quantitative convergence of individual core states would require higher cutoff energies, nevertheless, they offer a good qualitative insight. Two distinct features, depending on the velocity regime, are immediately noticeable. At low velocities the core occupied levels remain in their energy range, while the valence band shows that some dynamical states acquire energies that eventually reach hundreds of eV above the Fermi energy, forming an increasing set of ballistic electrons that would eventually be emitted from the sample. At high velocity the latter effect is more pronounced, both in the number of electrons and the energy scale. More importantly, we see an effect that is absent at low velocity, related to the excitation of core electrons of the projectile into valence band energies and further into the ballistic range. It is interesting to note that the oscillations in the energy expectation values of the states do not commensurate with lattice spacing, but change with velocity, rather maintaining a constant period in time. This indicates that the oscillations are intrinsic to the dynamical process rather than to the external periodicity. This behaviour could be seen as a flapping of the core electrons as shown in Fig. 3, with a dynamical re-shaping in real space illustrated in Fig. 4.
In summary, for Ni, like other transition metals that show a very high electronic stopping power, core electrons were found to have a major contribution in it, particularly those of the projectile. Adding explicit electrons in the simulation has the dual effect of adding more excitation channels, mainly in the form of electrons of the host, and making the ion potential deeper when ionization occurs, mainly in the projectile. While considering only 10 dynamical electrons per atom with frozen core could be a good approximation below of velocity, the 18 electron approximation is valid up to , before saturating. For larger velocities, more electrons need to be taken into account to reproduce a reasonable value for the stopping power; specially for the projectile ion including its core electron flapping behaviour.
We are thankful to T. Ogitsu for making the Ni26 pseudopotential available. R. U. and E. A. would like to acknowledge financial support from MINECO-Spain through Plan Nacional Grants No. FIS2012-37549 and FIS2015-64886, and FPI Ph.D. Fellowship Grant No. BES-2013-063728, along with the EU Grant “ElectronStopping” in the Marie Curie CIG Program. Work by R. U. (during visit hosted by A. A. C.) and by A. A. C. performed under the auspices of the US Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 and funded as a part of the Energy Dissipation to Defect Evolution (EDDE), an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Basic Energy Sciences.
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