5 Ultrafast Nonlinear Plasmonics
183
5.4 Time Dependent Metal Dielectric Function Change
Modification of the distribution f (or temperature) of the electrons in the conduction
band and subsequent lattice heating translate into modifications of the dielectric
function ε of the metal and thus of its optical absorption [26, 29, 52, 61, 65].
These can be computed provided the induced change ωε is connected to the electron
distribution change ω f (or ωT e ) and to the lattice temperature rise ωT L . This is done
analyzing their impacts on the interband (bound electrons) and intraband (quasi-free
electrons) contributions to ε (Eqs. 5.1 and 5.5). The dominant mechanisms involved
in modification of the former, ωε ib , and latter, ωε in , contributions are associated to
electron and lattice heating, respectively. ωε ib , thus usually dominates when most
of the excess energy is stored in the electrons, i.e., on a short time scale, while ωε in
is dominant on a long time-scale when energy has been transferred to the lattice,
i.e., after electrons-lattice thermalization (Fig. 5.4). These different contributions are
discussed in the following.
5.4.1 Electronic Contribution
The electronic contribution due to energy injection in the electrons is important only
if energy is in the electron gas. It mostly impacts the interband term ωε ib , with a
smaller intraband contribution due to the change of the electron scattering rates, both
effects thus decaying with electron cooling to the lattice (Fig. 5.4).
5.4.1.1 Interband Dielectric Function
Change of the interband contribution ωε ib is a consequence of modification of the
interband absorption spectrum due to electron distribution smearing (Fig. 5.3), [26,
29, 110]. Its calculation requires connection of ωε ib
2 (ω) to ω f , which can be done in
gold and silver, using the band structure models of Rosei and co-workers (Eq. 5.9). In
these models, the spectral width of the electronic states is neglected [122]. Estimation
of its impact on ωε ib is difficult, since its inclusion introduces an additional parameter
when comparing experimental and theoretical results, which could compensate for
deviation between the real and model band structures. As a first approximation we
have neglected it, an approximation justified by the good agreement between the
calculated and measured transient optical property spectra [26, 32, 52]. One has also
to keep in mind that the Rosei models are valid for frequencies ω close to ω ib and a
quasi-thermalized electron gas. For strongly nonequilibrium electrons and frequency
away from ω ib significant deviations can take place and have to be taken into account
to quantitatively compare the experimental and theoretical results. In this regime, the
simple model of a parabolic conduction band and undispersed d-bands can be used,
183
5.4 Time Dependent Metal Dielectric Function Change
Modification of the distribution f (or temperature) of the electrons in the conduction
band and subsequent lattice heating translate into modifications of the dielectric
function ε of the metal and thus of its optical absorption [26, 29, 52, 61, 65].
These can be computed provided the induced change ωε is connected to the electron
distribution change ω f (or ωT e ) and to the lattice temperature rise ωT L . This is done
analyzing their impacts on the interband (bound electrons) and intraband (quasi-free
electrons) contributions to ε (Eqs. 5.1 and 5.5). The dominant mechanisms involved
in modification of the former, ωε ib , and latter, ωε in , contributions are associated to
electron and lattice heating, respectively. ωε ib , thus usually dominates when most
of the excess energy is stored in the electrons, i.e., on a short time scale, while ωε in
is dominant on a long time-scale when energy has been transferred to the lattice,
i.e., after electrons-lattice thermalization (Fig. 5.4). These different contributions are
discussed in the following.
5.4.1 Electronic Contribution
The electronic contribution due to energy injection in the electrons is important only
if energy is in the electron gas. It mostly impacts the interband term ωε ib , with a
smaller intraband contribution due to the change of the electron scattering rates, both
effects thus decaying with electron cooling to the lattice (Fig. 5.4).
5.4.1.1 Interband Dielectric Function
Change of the interband contribution ωε ib is a consequence of modification of the
interband absorption spectrum due to electron distribution smearing (Fig. 5.3), [26,
29, 110]. Its calculation requires connection of ωε ib
2 (ω) to ω f , which can be done in
gold and silver, using the band structure models of Rosei and co-workers (Eq. 5.9). In
these models, the spectral width of the electronic states is neglected [122]. Estimation
of its impact on ωε ib is difficult, since its inclusion introduces an additional parameter
when comparing experimental and theoretical results, which could compensate for
deviation between the real and model band structures. As a first approximation we
have neglected it, an approximation justified by the good agreement between the
calculated and measured transient optical property spectra [26, 32, 52]. One has also
to keep in mind that the Rosei models are valid for frequencies ω close to ω ib and a
quasi-thermalized electron gas. For strongly nonequilibrium electrons and frequency
away from ω ib significant deviations can take place and have to be taken into account
to quantitatively compare the experimental and theoretical results. In this regime, the
simple model of a parabolic conduction band and undispersed d-bands can be used,