*4.2. Electron–Lattice Coupling*

Three models have been suggested for atomic displacement induced by electronic excitation: Coulomb explosion (CE) [3,4], thermal spike (TS) [50] and exciton model [30,95–97]. The neutralization time of the ionized region along the ion path is generally too short, and the fraction of the charged sputtered ions is small, e.g., 100 MeV Xe ions on SiO<sup>2</sup> glass [48]. Hence, the CE model is unsound. However, a small atomic separation during the short time might be enough for electron–lattice coupling (a key for electronic excitation effects), which will be discussed later. A crude estimation of the evaporation yield for SiO<sup>2</sup> based on the TS model appears to be far smaller than the experimental sputtering yield [55] and thus the TS model is also unsound. Moreover, the electron–lattice coupling or transfer mechanism of electronic energy into the lattice is not clear in the model. In the exciton model, the non-radiative decay of self-trapped excitons (STX, i.e., localized excited-state of electronic system coupled with lattice) leads to atomic displacement. According to the exciton model (or bandgap scheme), it is anticipated that the energy of the atoms in motion from the non-radiative decay of STX is comparable with the bandgap, leading to a larger sputtering yield with a larger bandgap, discarding the argument for the efficiency of STX generation from the electron–hole pairs, which is inversely proportional to the bandgap. This bandgap scheme is examined below. The effective depth contributing to the electronic sputtering of WO<sup>3</sup> has been obtained to be 40 nm, which is nearly independent of S<sup>e</sup> [98], which would shed light on understanding the electronic sputtering; therefore, more data are desired.

The electronic sputtering yield Ysp super-linearly depends on the electronic stopping power (Se), and Ysp at S<sup>e</sup> = 10 keV/nm is taken to be a representative value, which is plotted as a function of the bandgap (Eg) in Figure 11 from [56], including the present TiN result. The optical absorbance (defined as log10(Io/I), I<sup>o</sup> and I being the incident and transmitted photon intensities) of TiN films are measured, and the direct bandgap E<sup>g</sup> is obtained to be 4.5 eV for a film thickness of 25–50 nm, which decreases to 2.8 eV for a film thickness of ~180 nm by using the relation: (absorbance • photon energy)<sup>2</sup> is proportional to photon energy—Eg. The thickness dependence of E<sup>g</sup> is under investigation by considering the influence of the reflectivity, film growth conditions and experimental problems, such as stray light, etc. A large variation

has been reported for Eg, 4.0 eV (film thickness of 260 nm on Si substrate) (Popovic et al.) [99], 3.4 eV (thickness of ~100 nm on glass substrate) (Solovan et al.) [100] and 2.8–3.2 eV (film thickness of 460 nm on glass substrate) (Kavitha et al.) [101]. In this study, E<sup>g</sup> is taken to be 4 eV and this choice is tolerable in the following discussion. It has been reported that the bandgap is reduced by 0.06 eV under a 400 KeV Xe ion implantation at 10<sup>16</sup> cm−<sup>2</sup> [99]. High-energy ion impact effects on optical properties are under way. It can be observed that the bandgap scheme seems to work for E<sup>g</sup> > 3 eV [56]. A large deviation (two orders of magnitude) from the upper limit (dashed line indicated in Figure 11) is observed for ZrO2, MgO, MgAl2O<sup>4</sup> and Al2O3. The existence of STX is known for limited materials, rare gas solids, SiO<sup>2</sup> and alkali halides [30,95,96]. The STX does not exist for MgO and probably does not exist for Al2O<sup>3</sup> [102]. The deviation for MgO and Al2O<sup>3</sup> could be explained by the non-existence of STX. The numbers of electron–hole pairs leading to STX are inversely proportional to Eg, which could be a reason for the dependence of the sputtering yields for E<sup>g</sup> < 3 eV. In any case, the single parameter of the band gap is insufficient for the explanation of the bandgap dependence of the sputtering yields.

**Figure 11.** Sputtering yield at Se = 10 keV/nm vs. bandgap. Data from [56], TiN (present result) and LiF data from [62]. Dotted line is a guide for eyes (Eg > 3 eV).

Martin et al. [102] argued that STX exists for materials with small elastic constants. Following this suggestion, sputtering yields are plotted as a function of the elastic constant (C11) in Figure 12. Here, C<sup>11</sup> (GPa) is taken to be 87 (SiO2), 348 (SrTiO3), 497 (Al2O3), 294 (MgO), 270 (TiO2), 210 (ZnO), 299 (MgAl2O4), 242 (Fe2O3), 35 (KBr) and 114 (LiF) [85], and, for other materials, 403 (CeO2) [103], 224 (Y2O3) [104], 400 (ZrO2) [105], 13 (WO3) [106], 126 (Cu2O) [107], 135 (CuO) [108], 388 (Si3N4) as an average of the values [109,110], 345 (polycrystalline-AlN) [111], which is smaller by 16% than 410 (AlN single crystal) [112], 234 (Cu3N) [113], 500 (SiC) [114] and 625 (TiN) [115]. It can be observed for oxides (the most abundant data are available at present) that Ysp decreases exponentially with an increase in the elastic constant for C<sup>11</sup> < 300 GPa, except for MgO and ZrO2. Ysp for nitrides and SiC is larger than that for oxides at a given C11, and these are to be separately treated. It can be understood that the elastic constant represent the resistance of lattice deformation by electronic energy deposition. However, a single parameter, either the bandgap or elastic constant, is not adequate, and at least one more parameter is necessary. Furthermore, parameters other than those mentioned above are to be explored. More data for nitrides, alkali halides and especially carbides are desired.

**Figure 12.** Sputtering yield at Se = 10 keV/nm vs. elastic constant. Sputtering yield of TiN (present result), LiF from [62] and others from [56]. Dotted line is a guide for eyes for the most abundant available data of oxides (o, •).

Finally, a mechanism for the electron–lattice coupling is discussed. In an ionized region along the ion path, Coulomb repulsion leads to atomic motion, which is not adequate to cause sputtering because of its short neutralization time. Nevertheless, displacement comparable with the lattice vibration amplitude (one tenth of the average atomic separation, dav of ~0.25 nm for a-SiO2) is highly achievable during the neutralization time. As a first step, the time required for the Si+–O<sup>+</sup> displacement of 0.025 nm (one tenth of dav) from dav is estimated to be ~15 fs using a formula [116]. Also, the time is estimated to be ~ 15 fs and ~ 12 fs for the Zn+–O<sup>+</sup> displacement of 0.02 nm from dav of 0.2 nm in ZnO and for the Ti+–N<sup>+</sup> displacement of 0.02 nm from dav of 0.2 nm in TiN, respectively. A similar situation has been reported for the Fe+–O<sup>+</sup> displacement of 0.01 nm in Fe2O<sup>3</sup> (~7 fs) [60], the K+–Br<sup>+</sup> displacement of 0.01 nm in KBr (~9 fs) and the Si+–C<sup>+</sup> displacement of 0.01 nm in SiC (~6 fs). These suggest a possibility that a small displacement comparable with the lattice vibration amplitude caused by Coulomb repulsion during the short neutralization time leads to the generation of a highly excited-state coupled with the lattice (h-ESCL), and h-ESCL is considered to be equivalent to STX or multi STX. The non-radiative decay of h-ESCL leads to atomic displacement (a larger displacement results in sputtering and smaller displacement results in phonon generation or lattice distortion).
