Ab initio electronic stopping power and threshold effect of channeled slow light ions in ${\mathrm{HfO}}_2$

We present an ab initio study of the electronic stopping power of protons and helium ions in an insulating material, ${\mathrm{HfO}}_2$. The calculations are carried out in channeling conditions with different impact parameters by employing Ehrenfest dynamics and real-time, time-dependent density functional theory. The satisfactory comparison with available experiments demonstrates that this approach provides an accurate description of electronic stopping power. The velocity-proportional stopping power is predicted for protons and helium ions in the low-energy region, which conforms to the linear response theory. Due to the existence of a wide band gap, a threshold effect in the extremely low velocity regime below excitation is expected. For protons, the threshold velocity is observable, while it does not appear in the case of helium ions. This indicates the existence of extra energy-loss channels beyond the electron-hole pair excitation when helium ions are moving through the crystal. To analyze it, we checked the charge state of the moving projectiles and an explicit charge exchange behavior between the ions and host atoms was found. The missing threshold effect for helium ions is attributed to the charge transfer, which also contributes to energy loss of the ion.

Figure 1

Electronic stopping power for protons (a) and helium ions (b) versus velocity along the middle axes of $\langle 100\rangle$, $\langle 110\rangle$, and $\langle 111\rangle$ channels of ${\mathrm{HfO}}_2$. The lines are guides to the eye. The inset in (a) is enlargement of the main figure. The red curves are the best fit for stopping power in the $\langle 100\rangle$ channel.

Figure 2

Schematic drawing of the Auger capture process. An electron decays from a $|{\phi }_v\rangle$ state in the valence band to a bound state of the ion $|{\phi }_b\rangle$. The energies of the states respectively are ${\varepsilon }_v$ = $k_v^2$/2 and ${\varepsilon }_b$. An electronic excitation (either an electron-hole pair or a collective excitation) is created at the same time in the medium. The energy of the excitation is $\omega$ = ${\varepsilon }_v$ – ${\varepsilon }_b$.

Figure 3

Snapshots of time evolution of the electron density change caused by a ${\mathrm{He}}^{2+}$ ion moving through ${\mathrm{HfO}}_2$ crystal with $v$ = 1.0 a.u. along the middle axis of the $\langle 100\rangle$ channel (side view): (a) $t=0.059$ fs, the ion is above the crystal, (b) $t=0.095$ fs, the ion is entering the channel, (c) $t=0.486$ fs, the ion is penetrating along the channel, and (d) $t=0.628$ fs, the leaving ion retains some electrons after the collision. The gray regions are the change of electron distribution caused by the intruding ions.

Figure 4

Electrons induced by protons (a) and helium ions (b) versus $z$ coordinates in the low-velocity regime along the middle axis of the $\langle 100\rangle$ channel. The vertical dashed lines are the positions of O layers, and the vertical solid lines are the positions of Hf layers.

Figure 5

Electrons induced by 0.06-a.u. protons (a) and helium ions (b) versus $z$ coordinates along the middle axis of the $\langle 110\rangle$ channel. The vertical dash-dotted lines are positions of crystalline layers comprising both O and Hf atoms.

Figure 6

Electron occupation number distribution after the collision ends. The left side of the vertical lines shows the occupation of the ground state and the right side shows the excited states. The black curve and the red curve are cases of protons at 0.01 a.u. in the $\langle 110\rangle$ channel and 0.5 a.u. in the $\langle 100\rangle$ channel, respectively. See more details in the text.

Figure 7

Electronic stopping power of helium ions in ${\mathrm{HfO}}_2$ as a function of the velocity along two trajectories in the $\langle 100\rangle$ channel (a) and three trajectories in the $\langle 110\rangle$ channel (b). The lines are guides to the eye. The inset shows a sectional view of the $\langle 100\rangle$ channel and the trajectories. The gray circles and the blue ones represent host atoms in different transverse planes (defining the channel), while the black circles show the projectile positions for different impact parameters, where I.P. is the impact parameter. For the five trajectories impact parameters are 1.279 Å, 0.904 Å, 1.809 Å, 1.357 Å, and 0.904 Å, respectively.

Figure 8

Electrons induced by ${\mathrm{He}}^{2+}$ along trajectories with different impact parameters in the $\langle 100\rangle$ channel (a) and in the $\langle 110\rangle$ channel (b) for a given velocity of 0.3 a.u. The curves correspond to the trajectories in Fig. 7. The vertical dashed lines and solid lines in (a) are the positions of O layers and Hf layers, respectively. The vertical dash-dotted lines in (b) are positions of crystalline layers comprising both O and Hf atoms.

Figure 9

Electron densities (a) and axial force (b) exerted on helium ion for a given velocity of 0.3 a.u. versus $z$ coordinates in ${\mathrm{HfO}}_2$ along three trajectories in $\langle 110\rangle$. The curves correspond to trajectories in Fig. 7. The vertical dashed lines are positions of crystalline layers comprising both O and Hf atoms.