Measuring Electron Gas Relaxation in Gold through Second Harmonic Generation

Paul SanGiorgio
Advisor: Tom Donnelly

The very unusual properties of electrons in a metal lattice are responsible for many of the beneficial properties of metals. Instead of being tightly bound to single nuclei, electrons are very loosely bound and behave much like gases in a periodic potential well. At a well-defined temperature, t, the physics of the electron gas is very well understood. The number of electrons with energy, E, is given by the Fermi-Dirac distribution:

                                        N(E) = (1 + exp[(E-Ef)/kt])-1* D(E)*dE

where D(E) is the density of states function, which is experimentally easy to determine, but theoretically impossible to calculate for all but the simplest models. Likewise, Ef, the Fermi energy is equally difficult to calculate theoretically, but experimentally a snap. This electron distribution is responsible for many physical effects, including the photoelectric effect, the high electric conductivity of metals, and also the high reflectivity of metals.

Yet, this distribution is only valid for a well-defined t and therefore only in equilibrium situations. In most applications this is not a limitation since the thermalization time in most metals is on the order of 1 ps, and until recently our temporal resolution was limited to 0.5 - 1 ps. But, with the development of ultra-fast lasers, which produce light pulses 15 fs long (0.015 ps), we are able to characterize properties on much smaller time scales.

One metallic property that is particularly sensitive on these time scales is the surface generation of second harmonic light. In a SHG event two photons of frequency w are absorbed by an electron and a single photon of frequency 2w is emitted soon after. This is a very infrequent effect (the intensity of the SHG light is typically ten or eleven orders of magnitude less than the incident intensity) but it is easily detectable by a sensitive photo multiplier tube. A first approximation on the intensity of SHG intensity gives the relationship:

                                        I(2w) is proportional to I(w)2

which is valid for low intensities ( < 10 GW/cm2), even on femtosecond time scales. Yet, at higher intensities, a supraquadratic behavior is observed, which has been commonly studied, but generally not well understood. Recently, using femtosecond techniques, K Moore and T Donnelly proposed a model for SHG which attributes the behavior to nonequilibrium electron distributions on the gold surface. The agreement with the experimental data gathered to the predictions of the model are excellent, considering that there are effectively no free parameters in the model.

The purpose of my work is to study and also to predict how the SHG changes as the electron distribution relaxes. This is done through adding pump-probe excitation to the setup used by K Moore and T Donnelly. Unlike in their previous work, however, the thermalization of the electron distribution will not be negligible, since the time scale I will be studying is the 0 - 10 ps, in approximately 25 fs steps. Thus, a reasonable model for electron thermalization is necessary.

Electrons thermalize with a metal lattice through two effects: electron-electron scattering and electron-phonon scattering. The electron-electron scattering causes the electrons to relax to a FD distribution within typically 500 fs, and the thermalized distribution interacts with the metal lattice through electron-phonon scattering which takes approximately 5 ps. Previous models for understanding this relaxation have many shortcomings, most of which can be avoided thanks to the recent availability of computing power. My model simulates the individual collisions between electrons using very simple rules - essentially using only the Pauli exclusion principle and conservation of energy - and also similar rules for phonon scattering. The preliminary results of the model have been good, but numerous questions remain. Through research and simulations, I will create a functional model, which will then be tested in the lab. There is still much to be done in the lab, yet we are optimistic that we will have usable data by fall break. Even if experimental data is never collected, the model is an interesting bit of physics in its own right.

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| Math Department | Last modified: September 2000