7 Ultrafast and Nonlinear Plasmon Dynamics
249
7.2 Damping of Surface Plasmon Polaritons
7.2.1 Theory of Radiative and Nonradiative Decay
The coherent electronic excitation of a medium is followed by fast electronic dephasing and the subsequent absorption and decay of the polarization into electron
hole pairs. 8 Here we will discuss the radiative and non-radiative relaxation dynamics of SPPs as the fastest initial processes describing the light-matter interaction. We
will restrict the discussion to the homogeneous SPP response, i.e., in the absence of
ensemble effects and different inhomogeneities. We will discuss the basic physics
of plasmon dephasing in this section, followed in subsequent sections by different
frequency- and time-domain experimental linear and nonlinear spectroscopic results
for its experimental determination.
Of primary interest is the electronic dephasing, that is, the eventual loss in phase
coherence of the collective and initially phase coherent oscillation of the free electron
gas (plasma oscillation). In contrast to semiconductors, which allow for a low and
variable carrier density through controlled doping, the carrier density in metals is
comparatively high and fixed (Table 7.1). Those high carrier densities immediately
imply a high scattering and thus high dephasing rate. The SPP decoherence time is
therefore fundamentally linked to the effective relaxation time in the Drude dielectric function as the response function that determines the temporal evolution of the
induced optical polarization in response to an applied optical field. Consequently, to
first order, the Drude relaxation time τ D sets an upper limit for the dephasing time
T 2 for a localized SPP.
The macroscopic optical response of metals in general, including the SPP resonance for plasmonic metal nanostructures, reflects the underlying elementary electron
dynamics of the bound and conduction electrons involved. Specifically, the linewidth
and shape of the SPP resonance in the frequency domain, or its Fourier transform
in the form of the free-induction decay in the time domain, describes the loss in
phase coherence, which in turn is directly linked to the dielectric function. In the following we discuss the ultrafast electron dynamics of spherical and spheroidal metal
nanoparticles as model systems using analytical treatments. The results can readily
be generalized for more complex geometries using numerical techniques. For small
enough particles (R √ λ) the excitation is dominated by the dipolar SPP response,
with polarizability given by Eq. 7.12 for a sphere or Eq. 7.13 for a spheroid.
8 Following the typically up to 10s fs coherent evolution of electronic excitations, different processes
govern the incoherent carrier cooling and equilibration. The decay of the coherent excitation into
electron-hole pairs gives rise to hot non-equilibrium and non-thermal carrier distributions. Electron–
electron scattering leads to thermalization of the hot electrons within at most a few hundred fs and
can often be described by the Fermi liquid theory. Electron–phonon interaction on 100 fs to ps time
scales leads to the subsequent equilibration with the lattice degrees of freedom. Although not the
subject of this review, these processes lead to transient variations of the dielectric function and its
frequency dependence, ultimately due to the deposited energy in the form of heat. The processes
need to be considered in time resolved experiments, especially with high pump intensities and large
excitation densities, giving rise to a nonlinear response.
249
7.2 Damping of Surface Plasmon Polaritons
7.2.1 Theory of Radiative and Nonradiative Decay
The coherent electronic excitation of a medium is followed by fast electronic dephasing and the subsequent absorption and decay of the polarization into electron
hole pairs. 8 Here we will discuss the radiative and non-radiative relaxation dynamics of SPPs as the fastest initial processes describing the light-matter interaction. We
will restrict the discussion to the homogeneous SPP response, i.e., in the absence of
ensemble effects and different inhomogeneities. We will discuss the basic physics
of plasmon dephasing in this section, followed in subsequent sections by different
frequency- and time-domain experimental linear and nonlinear spectroscopic results
for its experimental determination.
Of primary interest is the electronic dephasing, that is, the eventual loss in phase
coherence of the collective and initially phase coherent oscillation of the free electron
gas (plasma oscillation). In contrast to semiconductors, which allow for a low and
variable carrier density through controlled doping, the carrier density in metals is
comparatively high and fixed (Table 7.1). Those high carrier densities immediately
imply a high scattering and thus high dephasing rate. The SPP decoherence time is
therefore fundamentally linked to the effective relaxation time in the Drude dielectric function as the response function that determines the temporal evolution of the
induced optical polarization in response to an applied optical field. Consequently, to
first order, the Drude relaxation time τ D sets an upper limit for the dephasing time
T 2 for a localized SPP.
The macroscopic optical response of metals in general, including the SPP resonance for plasmonic metal nanostructures, reflects the underlying elementary electron
dynamics of the bound and conduction electrons involved. Specifically, the linewidth
and shape of the SPP resonance in the frequency domain, or its Fourier transform
in the form of the free-induction decay in the time domain, describes the loss in
phase coherence, which in turn is directly linked to the dielectric function. In the following we discuss the ultrafast electron dynamics of spherical and spheroidal metal
nanoparticles as model systems using analytical treatments. The results can readily
be generalized for more complex geometries using numerical techniques. For small
enough particles (R √ λ) the excitation is dominated by the dipolar SPP response,
with polarizability given by Eq. 7.12 for a sphere or Eq. 7.13 for a spheroid.
8 Following the typically up to 10s fs coherent evolution of electronic excitations, different processes
govern the incoherent carrier cooling and equilibration. The decay of the coherent excitation into
electron-hole pairs gives rise to hot non-equilibrium and non-thermal carrier distributions. Electron–
electron scattering leads to thermalization of the hot electrons within at most a few hundred fs and
can often be described by the Fermi liquid theory. Electron–phonon interaction on 100 fs to ps time
scales leads to the subsequent equilibration with the lattice degrees of freedom. Although not the
subject of this review, these processes lead to transient variations of the dielectric function and its
frequency dependence, ultimately due to the deposited energy in the form of heat. The processes
need to be considered in time resolved experiments, especially with high pump intensities and large
excitation densities, giving rise to a nonlinear response.