Quantum hydrodynamic model for the nonlinear electron dynamics in thin metal films

A quantum hydrodynamic (fluid) model, derived from the Wigner-Poisson equations, is used to investigate the ultrafast electron dynamics in thin metal films. The hydrodynamic equations, which include exchange and correlation effects, can be combined into a single nonlinear Schrödinger-type equation. The fluid model is first benchmarked against a density-functional calculation for the ground state, with good agreement between the two approaches. The ultrafast nonlinear electron dynamics is then investigated and compared to recent semiclassical results obtained with a Vlasov-Poisson approach.

Figure 1

Comparison between the ground states computed from the DFT equations (solid line) and with the QHD approach (dashed line) for a film of thickness $L=50L_F$ and $\sigma =L_F$. (a) Electron density normalized to the background density $n_0$ (b) Profile of the Hartree potential $\left(V_H\right)$ and effective potential $\left(V_{\text{eff}}\right)$ normalized to the Fermi energy.

Figure 2

Same as Fig. 1 for $\sigma =2L_F$.

Figure 3

Ground-state properties: Thomas-Fermi energy (a) and effective potential energy (b) against $\sigma /L_F$ computed using the DFT equations and the QHD approach for a film of thickness $L=50L_F$. Energies are normalized to the Fermi energy.

Figure 4

QHD model. Electric dipole for $L=50L_F$ and $\sigma =2L_F$. Left column: Full time history on a logarithmic scale; middle column: Zoom for short times (the dashed straight line is an exponential with damping rate $\gamma =0.055{\omega }_{\text{pe}}$); right column: Frequency spectrum. From top to bottom: $H=0.9405$, $H=0.5$, $H=0.1$.

Figure 5

Electric dipole from the Vlasov simulations for $L=50L_F$ and $\sigma =2L_F$. (a) Full time history on a logarithmic scale. (b) Zoom for short times. (c) Frequency spectrum. The straight line is an exponential with damping rate ${\gamma }_{\text{Vlas}}=0.035{\omega }_{\text{pe}}$.

Figure 6

QHD model. Time evolution of the potential energy (left column), center-of-mass energy (middle column), and thermal energy (right column) for $L=50L_F$, $\sigma =2L_F$. From top to bottom: $H=0.9405$, $H=0.5$, $H=0.1$. Energies are expressed in units of the Fermi energy.

Figure 7

QHD model. Time evolution of the potential energy (left column), center-of-mass energy (middle column), and thermal energy (right column) for $L=100L_F$, $\sigma =2L_F$. Top panels: $H=0.9405$; bottom panels: $H=0.1$. Energies are expressed in units of $E_F$.

Figure 8

Time evolution of the potential, center-of-mass and thermal energies for the Vlasov-Poisson model. (a) $L=50L_F$. (b) $L=100L_F$. For both cases, $\sigma =2L_F$.