Strong-field photoemission from surfaces: Theoretical approaches

The problem of highly nonlinear photoemission from a metal surface is considered using analytical and numerical approaches. Descriptions are found which cover both the weak-field and the strong-field regimes and the transition between them. The results of a time-dependent perturbation theory are in very good agreement with those from more numerically involved schemes, including a variational version of the Floquet method and a Crank-Nicolson-like numerical scheme. The implemented Crank-Nicolson variant uses transparent boundary conditions and an incident plane-wave state in the metal. Both numerical approaches give very similar results for weak and intermediate fields, while in the strong-field regime the Crank-Nicolson scheme is more effective than the Floquet method. We find an enhancement in the effective nonlinearity in the weak-field regime, which is caused by surface scattering of the final state. The presented theory also covers angular emission probabilities as a function of light intensity and explains an increase toward forward emission with growing field strength.

Figure 1

Sketch of the plane-wave components of the Green's function (perturbation at $s^{\prime }$), taking into account a finite reflection amplitude at the metal-vacuum interface.

Figure 2

The reflection coefficients as functions of momentum $p_n$ from analytical [dashed red: $R^{\left(1\right)}$ only; solid green: $R^{\left(1\right)}{R}^{\left(2\right)}$] and numerical (dotted blue) computations for different field strengths (a) $F=0.2$ V/nm, (b) 3 V/nm, (c) 10 V/nm, and (d) 30 V/nm.

Figure 3

Tunneling probability for a triangular barrier as a function of the static electric field: exact result (cyan line), WKB approximation (red dashed line), and Crank-Nicolson scheme (triangles). Work function and Fermi energy are $5.5$ and $4.5$ eV, respectively.

Figure 4

Emission probability, defined as the ratio of (time-averaged) outgoing and incoming current densities, as a function of the peak electric field of the incident light: results of continuous-wave Crank-Nicolson computations with the delocalized initial-state (solid green) and perturbative approach [Eq. (22)] (dotted orange). Results of the variational approach showing the contributions from four-photon (dashed blue) and five-photon (dot-dashed blue) absorption. Work function and initial-state energy (Fermi energy) are 5.5 and 4.5 eV, respectively.

Figure 5

(a) Normalized time-resolved surface current as a function of both the peak electric field and temporal coordinate. Dashed line indicates electric force. Work function and Fermi energy are $5.5$ and $4.5$ eV, respectively. Lower panel: exemplary curves for Keldysh parameters (b) $\gamma =6$, (c) $\gamma =3.6$, (d) $\gamma =0.6$, and (e) $\gamma =0.3$.

Figure 6

Magnitude of [(a), (b)] wave function $\left|\psi \right(z,t\left)\right|$ and [(c), (d)] current density as a function of time and space. Dashed lines indicate electric force for reference. Solid white lines show corresponding classical trajectories. Parameters: $W=5.5$ eV and $E_F=4.5$ eV.

Figure 7

(a) Crank-Nicolson computations for photoemission as a function of the peak electric field $F_0$ for single- (dashed) and two-cycle (solid) pulses and tunneling approximation (dot-dashed cyan). (b) Comparison of Crank-Nicolson photoemission yield for localized (dashed) and delocalized (solid) initial-state computations (the dashed curve is fit to agree with the solid curve for weak fields). The inset schematically depicts the metal-vacuum interface (spatial coordinate $z=0$) for (c) localized and (d) delocalized initial states.

Figure 8

Normalized angular distributions of emitted electrons in the low-temperature limit [Eq. (32)] for field strengths of $F=1$ V/nm ($\gamma =18$, dotted), $F=12$ V/nm ($\gamma =1.5$, dashed), and $F=30$ V/nm ($\gamma =0.6$, solid), derived from perturbation theory. The angles denote the emission direction perpendicular to the metal surface.

Figure 9

(a)–(c) Emission probability and effective nonlinearity computed with the Floquet method. Four-photon contribution (dashed), five-photon contribution (dot-dashed), and total emission probability (solid) for work functions of (a) $W=5.2$ eV and (b) $W=5.5$ eV. (c) Effective nonlinearity for $W=5.2$ eV (dot-dashed), $W=5.5$ eV (solid), and $W=7.1$ eV (dashed). (d)–(e) perturbation-theory computations, illustrating the role of reflection in the enhancement of the effective nonlinearity ($W=5.5$ eV). (d) Effective nonlinearity with (solid) and without (dashed) inclusion of surface reflection. (e) Field-dependent reflection coefficients (four-photon process) from Eqs. (14).