Hydrodynamic approximation for the nonlinear response of a metal surface

We present semi-classical and quantized hydrodynamic models to obtain the quadratic electronic response of a plane-bounded electron gas. Explicit expressions for the dynamic image potential experienced by charged particles moving near a jellium surface are derived, up to third order in the projectile charge. These expressions are employed to compute the image potential at all distances outside the surface. Though nonlinear corrections are found to be more important far inside the solid than outside, our results indicate that the nonlinear image potential is enhanced with respect to the linear image potential by a factor that is for Al as large as $\sim 1.15$ near the surface in the case of a stationary particle ($v\to 0$) with positive unit charge $e$.