Ultrafast electron dynamics in metals: Real-time analysis of a reflected light field using photoelectrons

We propose an approach to address ultrafast charge-carrier dynamics of metals by analyzing the momentum change in photoelectrons interacting with a transient optical grating at a metal surface. Photoelectrons are excited by an ultraviolet femtosecond laser pulse which precedes an infrared pulse setting up the transient grating. We measure the kinetic energy of the photoelectrons which are accelerated by the grating’s ponderomotive potential and thus sample the respective electric field. The method is capable to access phase and amplitude differences between the incoming and the reflected light fields. The latter is determined by the response of the conduction-band electrons to the light field. We report on a demonstration of such an experimental scheme using a Gd(0001) surface and calculate the reflected field by a simplified transport equation. We derive a method to determine the average electron-scattering rate and study the time-dependent evolution of amplitude and phase of the reflected electric field including memory effects in the optically induced polarization dynamics. Finally, we discuss the required steps of this approach to probe ultrafast dynamics in metals experimentally.

Figure 1

(a) Acceleration of charged particles in the ponderomotive potential ${\varphi }_p$; (b)–(d) depict the experimental scheme of an infrared femtosecond laser pulse reflected at a metal surface. Incident and reflected parts overlap in space and form a two-dimensional transient grating. The electric field intensity is given by the gray scale. The intensity is maximum when the pulse center is located at the surface (c). Photoelectrons sense the standing wave and are analyzed in normal emission by an electron time-of-flight (TOF) spectrometer.

Figure 2

Top: photoemission spectrum of Gd(0001) with $h\nu =6\text{eV}$ taken at a temperature of 30 K. The inset includes a Gaussian fit of the surface-state peak. Bottom: circles indicate the kinetic energy of the peak measured for different delay, i.e., distance to the surface $x$ (top or bottom axis). The solid line is calculated by Eq. (3). Errors bars are within the symbol size. The oscillations are observed up to 17 ps; however for clarity $\Delta t<8\text{ps}$ is shown only.

Figure 3

Time-dependent variation in the photoemission intensity for different kinetic energies as indicated $\left(h{\nu }_{\text{UV}}=6\text{eV}\right)$. The amplitude variations scale with each other and depend on the gradient of the intensity with kinetic energy. Solid lines represent fits to the data by a sine function.

Figure 4

Top: photoemission spectra at indicated photon energies while the IR pulse is present. The intensity increase recognized for the 6 eV data at low energy was absent in Fig. 2 and is attributed to hot electrons generated by absorption of IR photons. Bottom: dispersion of the oscillation frequency $\Omega /2\pi$ with kinetic energy. The solid line is a fit by a square-root dependence of oscillation frequency on the kinetic energy.

Figure 5

[(a) and (d)] Incident fields with pulse duration $\tau$ of (a) 50 and (d) 1 fs. Respective reflected fields are given for [(b) and (e)] $\Gamma =1{\text{fs}}^{-1}$ and [(c) and (f)] $\Gamma =5{\text{fs}}^{-1}$.

Figure 6

[(a) and (c)] Calculated and (b) measured change in the electron momentum along the surface normal as a function of pump-probe delay, i.e., distance from the surface (cf. Fig. 2). Parameters of the calculation are $\omega =2.3{\text{fs}}^{-1}$ and $\Gamma =1{\text{fs}}^{-1}$. The pulse durations are chosen as 50 and 1 fs for panels (a) and (c), respectively. The measured momentum change depicted in panel (b) is based on the data shown in Fig. 2. The kinetic-energy modulation is converted into the momentum change using $\Delta p_x=\Delta E_{\text{kin}}\sqrt{\frac{m}{2E_{\text{kin}}}}$ with $E_{\text{kin}}=2.4464\text{eV}$.

Figure 7

Plots of the envelope function of $\Delta p_x$ as a function time delay for different pulse durations of the incident field $\tau$ and scattering rates $\Gamma$ as indicated. Note the different scalings of panels (a)–(c).

Figure 8

Plots of the envelope function of $\Delta p_x$ as a function of time delay and scattering rate $\Gamma$. On the connection line between the two bifurcations—left and right—the momentum change presents only saddle points. The solid line describes the position of the minimum and the dashed line is the position of maximum momentum change. Panels (a)–(c) describe the behavior for 100, 10, and 1 fs pulse duration, respectively.

Figure 9

Dependence of the phase $\alpha \left(\Delta t\right)$ in units of $\pi /2$ as a function of $\Gamma$ in ${\text{fs}}^{-1}$ for different pulse durations $\tau$ and time delay $\Delta t$ as indicated.