Suppressed and enhanced tunneling ionization of transition-metal atoms and cations: A time-dependent density-functional-theory study on nickel

We study the tunneling ionization (TI) of Ni, ${\mathrm{Ni}}^{+}$, and ${\mathrm{Ni}}^{2+}$ with a time-dependent density-functional-theory method and reproduce the puzzling suppression of the TI of Ni and ${\mathrm{Ni}}^{+}$ and the enhancement of TI in ${\mathrm{Ni}}^{2+}$. Numerical results reveal that for all three species the electron tunnels from a $4s$ orbital; that is, excitation precedes tunneling for both of the cations, for which the highest orbitals are $3d$. The effective radial potentials for the $d$ orbitals have a centrifugal barrier, while there is no such barrier for the $s$ orbitals. At the classical turning point for the $3d$ orbital, the $3d$ to $4s$ excitation energy is lower than the centrifugal potential for the $d$ orbitals. Two factors of opposite nature are identified in this work. On the one hand, electrons moving away from the nucleus in the intense laser fields induce an attractive potential that effectively lowers the energy level and thus suppresses tunneling. Excitation, on the other hand, has the opposite effect and enhances tunneling. The energy gap between $4s$ and $3d$ is small for ${\mathrm{Ni}}^{+}$ and therefore suppression wins. As the charge of the cation increases, the excitation energy becomes much greater, and for ${\mathrm{Ni}}^{2+}$ enhancement dominates. Based on a similar analysis, we expect enhanced TI for several transition-metal cations of charge 2 and higher.

Figure 1

Ionization probabilities of a neutral nickel atom and cations of nickel in 1500-nm, 90-fs laser pulses. The probabilities are presented in both the linear scale (top) and log scale (bottom). The experimental values are reproduced from a linearly scaled figure in Ref. [10]. Limited by the resolution of the print in [10], the probability reading near 0.01 may be far off the original value.

Figure 2

The induced potential $\mathrm{\Delta }{v}_{\downarrow }$ of Ni in $2\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$, of ${\mathrm{Ni}}^{+}$ in ${10}^{14}\mathrm{W}/{\mathrm{cm}}^2$, and of ${\mathrm{Ni}}^{2+}$ in $2.35\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$, 1500-nm lasers. The potential is obtained by solving the TDDFT equations for a time near 45 fs, when $E\left(t\right)=F$.

Figure 3

Energy diagram for Ni in a 1500-nm laser of $2\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$ (left) and for ${\mathrm{Ni}}^{+}$ in a 1500-nm laser of ${10}^{14}\mathrm{W}/{\mathrm{cm}}^2$ (right). Solid black line: the field-free energy level for the highest occupied spin orbital. Black dotted line: the potential of the electric field at the peak intensity plus the field-free electronic potential. Red solid line: the shifted energy level of $4s_{\downarrow }$ near $t=45$ fs when the electric field is maximized. Red dashed line: the TDDFT electronic potential at the same time. Green dot-dashed line: the unshifted and unoccupied $4s_{\downarrow }$ of ${\mathrm{Ni}}^{+}$. Blue dot-dashed line: the shifted $3d_{\downarrow }$ of ${\mathrm{Ni}}^{+}$ at the same time.

Figure 4

Energy diagram for ${\mathrm{Ni}}^{2+}$ in a 1500-nm laser of $2.35\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$. Solid black line: the field-free energy level for the highest occupied spin orbital, $3d_{\downarrow }$. Black dotted line: the potential of the electric field at the peak intensity plus the field-free electronic potential. Red solid line: the shifted energy level of $4s_{\downarrow }$ at the moment when the electric field is maximized. Red dashed line: the TDDFT calculated electronic potential at the peak intensity. Green dot-dashed line: the unshifted and unoccupied $4s_{\downarrow }$ of ${\mathrm{Ni}}^{2+}$.