Many-body theory of the neutralization of strontium ions on gold surfaces

Motivated by experimental evidence for mixed-valence correlations affecting the neutralization of strontium ions on gold surfaces, we set up an Anderson-Newns model for the Sr:Au system and calculate the neutralization probability $\alpha$ as a function of temperature. We employ quantum-kinetic equations for the projectile Green functions in the $\mathrm{finite}\text{−}U$ noncrossing approximation. Our results for $\alpha$ agree reasonably well with the experimental data as far as the overall order of magnitude is concerned, showing in particular the correlation-induced enhancement of $\alpha$. The experimentally found nonmonotonous temperature dependence, however, could not be reproduced. Instead of an initially increasing and then decreasing $\alpha$, we find over the whole temperature range only a weak negative temperature dependence. It arises, however, clearly from a mixed-valence resonance in the projectile's spectral density and thus supports qualitatively the interpretation of the experimental data in terms of a mixed-valence scenario.

Figure 1

Illustration of the time-dependent charge-transferring scattering of a ${\mathrm{Sr}}^{+}$ ion on a gold surface. Far from the surface, the two ionization energies ${\varepsilon }_0\left(t\right)$ and ${\varepsilon }_U\left(t\right)$ are equal to the first two ionization energies of a strontium atom and represent the projectile's $5s^1$ and $5s^2$ configurations. They shift upward due to the image interaction of the projectile with the surface. For simplicity, not shown is the hybridization between the projectile and surface states as the projectile closes its distance to the surface which is idealized by a step potential whose depth is the sum of the work function $\Phi >0$ and the Fermi energy $E_F>0$. The energies are on scale and the points indicated along the trajectory $z\left(t\right)$ are $z_{\mathrm{TP}}$, the turning point, and $z_c$, the point where the first ionization level crosses the Fermi energy.

Figure 2

Widths of the first $\left({\varepsilon }_U\right)$ and second $\left({\varepsilon }_0\right)$ ionization levels of a strontium projectile approaching a gold surface as computed from Eq. (11) on the basis of Hartree-Fock wave functions for the projectile and simple step-potential wave functions for the target. Atomic units are used, that is, energy is measured in hartrees and length in Bohr radii. Data are shown only for distances larger than 5 Bohr radii, which is the turning point of the collision trajectory. The width of a rubidium $5s$ level is also shown and contrasted with the width obtained for that level by Nordlander and Tully using the complex-scaling approach [56]. Notice, in contrast to Nordlander and Tully's Eq. (4.1) [56], our widths (11) contain a factor $2\pi$ and not a factor $\pi$. For the comparison, we corrected for this difference.

Figure 3

Coleman's pseudoparticle representation for the strontium projectile. Shown are the occupancies of the two ionization levels ${\varepsilon }_0$ and ${\varepsilon }_U$. The two single-occupied configurations $n_{p_{\sigma }}$ contain an electron only in the second ionization level. In the double-occupied state $n_d$ both ionization levels are occupied by electrons with opposite spin, whereas in the empty configuration none of the ionization levels are occupied.

Figure 4

Self-energies in the noncrossing approximation. Wavy, dashed, and double-dashed lines denote, respectively, fully dressed propagators for the empty (e), the single-occupied (p), and the double-occupied (d) configurations. The solid line is the bare Green function for the electrons of the surface. Starting at the left upper corner and proceeding clockwise, the diagrams denote, respectively, the self-energies ${\Sigma }_{0,\sigma } {\Sigma }_{U,\sigma },{\Pi }_d$, and ${\Pi }_e$ for the Green functions $P_{\sigma },D$, and $E$.

Figure 5

Sketch of the numerical scheme used to solve the double-time equations of motion (19, 20, 21, 22, 23, 24). The triangle marks the region in the two-dimensional time grid in which the entries of retarded Green functions have to be known in order to calculate retarded Green functions at the point indicated by the bullet. Likewise, the rectangle marks the region in which the entries of (retarded and less-than) Green functions are required in order to compute less-than Green functions at the point indicated by the bullet.

Figure 6

The constraints $R_0$ and $R_U$ as a function of $z$ for the Sr:Au system investigated by He and Yarmoff [30, 31]. The approximations reducing the double-time quantum kinetics of the Dyson equations to either a set of simple or generalized master equations (see Appendix) are valid only for $R_{0,U}\ll 1$. Hence, for ${\varepsilon }_0$ master equations could be used for $z>6$. But, for ${\varepsilon }_U$ master equations break down for almost the whole trajectory except for the narrow interval around $z\approx 9$ where the vanishing of the numerator in Eq. (36) leads to small values for $R_U$. The peak in $R_U$ around $z\approx 12$ signals the point where ${\varepsilon }_U$ crosses the Fermi energy.

Figure 7

Instantaneous physical quantities along the trajectory. The strontium projectile starts on the left as an ion at a distance $z=20$ with a velocity $v=0.0134$, reaches the turning point at $z_{\mathrm{TP}}=5$, and approaches $z=20$ again thereafter on the right. (a) Energy-level diagram. Both levels (solid lines) are broadened according to $\varepsilon \pm \Gamma$ (dashed lines) with the instantaneous $\Gamma$ shown in Fig. 2. (b) Occurrence probabilities at $T_s=400$ K for ${\mathrm{Sr}}^{+}$ (dashed and dashed-dotted lines), ${\mathrm{Sr}}^0$ (solid line), and ${\mathrm{Sr}}^{2+}$ (dotted line) as obtained from the $\mathrm{finite}\text{−}U$ model. The neutralization probability in this case is $\alpha =n_d\left(20\right)=0.185$. (c) Occurrence probability of ${\mathrm{Sr}}^0$ as obtained from the uncorrelated $U=0$ model which keeps only the first ionization level, that is, the upper onsite energy ${\varepsilon }_U$. In this case the $\alpha =0.01$. For other surface temperatures $T_s$ the results look similar.

Figure 8

Temperature dependence of the neutralization probability $\alpha =n_d\left(20\right)$ for a ${\mathrm{Sr}}^{+}$ ion hitting with $v=0.0134$ a gold surface. The turning point $z_{\mathrm{TP}}=5$. Also shown are the data of He and Yarmoff [31]. The solid and long-dashed lines are for the $\mathrm{finite}\text{−}U$ and the uncorrelated $U=0$ model, respectively, showing that correlations enhance the neutralization probability to the experimental order of magnitude. By moving the turning point farther away from the surface, we could make the results for $U\ne 0$ to overlap with the experimental data. However, we do not use $z_{\mathrm{TP}}$ as a fit parameter for reasons explained in the main text. The nonmonotonous temperature dependence of the experimental data cannot be reproduced regardless of the value of the turning point. The dashed-dotted and the dotted lines are the neutralization probabilities arising, respectively, from the numerical solution of the set of simple or the set of generalized master equations given in the Appendix.

Figure 9

Projectile spectral densities summed over the two spin orientations, respectively, at $z=5.0,5.5,6.0,6.5,7.0$, and $7.5$ on the outgoing branch of the trajectory [panels (a) to (f)] for a ${\mathrm{Sr}}^{+}$ ion hitting with $v=0.0134$ a gold surface at temperature $T_s=400$ K. For other surface temperatures in the range relevant for the experiment, the spectral functions look qualitatively similar. The black lines correspond to the instantaneous spectral densities at the given positions with the solid lines denoting the occupied and the dashed lines the total spectral densities. The vertical dotted lines indicate the instantaneous positions of the onsite energies ${\varepsilon }_0$ and ${\varepsilon }_U$ while the orange lines give the equilibrated occupied (solid lines) and total (dashed lines) spectral densities at the corresponding positions. The target's Fermi energy is located at $\omega =0$.