Electronic Stopping Power in Gold: The Role of $d$ Electrons and the $\mathbf{H}/\mathrm{He}$ Anomaly

The electronic stopping power of H and He moving through gold is obtained to high accuracy using time-evolving density-functional theory, thereby bringing usual first principles accuracies into this kind of strongly coupled, continuum nonadiabatic processes in condensed matter. The two key unexplained features of what observed experimentally have been reproduced and understood: (i) The nonlinear behavior of stopping power versus velocity is a gradual crossover as excitations tail into the $d$-electron spectrum; and (ii) the low-velocity $\mathrm{H}/\mathrm{He}$ anomaly (the relative stopping powers are contrary to established theory) is explained by the substantial involvement of the $d$ electrons in the screening of the projectile even at the lowest velocities where the energy loss is generated by $s$-like electron-hole pair formation only.

Figure 1

Electronic stopping power of H and He projectiles in gold as a function of projectile velocity. Results of our simulations are compared with the experimental data from Refs. 2, 3, 4 on single and polycrystalline thin gold films.

Figure 2

Excitation distribution $P(E)$ when H passes through gold with velocity 0.05 a.u., for time values between $t=0.1$ and 1.1 fs in steps of $\Delta t=0.1\mathrm{fs}$ (light color; larger amplitude for longer $t$; Gaussian broadening $\sigma =0.2\mathrm{eV}$). The dark curve is the electronic density of states $g(E)$ ($\sigma =0.5\mathrm{eV}$). $P(E)$ and $g(E)$ are in different scales.

Figure 3

Up: Excitation distribution $P(E)$ due to the passage of a H (top) or He projectile (bottom) in gold evaluated at $t=0.25\mathrm{fs}$ for various projectile velocities, $v=0.05–0.50\mathrm{a}.\mathrm{u}.$ in steps of 0.05 a.u. Increased projectile velocity gives curve with larger amplitude (indicated by arrows). The dashed and dotted vertical lines show the upper edge of the gold’s $5d$ band $E_d$ and the Fermi energy $E_F$. Down: Number of empty states below and above $E_d$, $N_1$ and $N_2$, and fraction $N_1/(N_1+N_2)$ versus projectile velocity, due to the excitations for H or He.

Figure 4

Up: The curve shows $R=S_e^{\mathrm{He}}/S_e^{\mathrm{H}}$ for jellium versus the electron density parameter $r_s$ [5]. The values of $R$ obtained for Au for $v=0.1$ and 0.5 a.u. are associated with $r_s=3.04$ and 1.49 a.u, respectively, following Ref. 4. The calculated ratio $R$ for a system made of Na atoms in bulk Au positions, $r_s=3.04$, is also presented. Down: Projection of KS states for the system with projectile onto the KS states of bulk Au, for H and He, subtracting the Au density of states.