250
M. B. Raschke et al.
From the calculated spectra for a Au sphere or spheroids with different aspect
ratios as shown in Fig. 7.5b, the corresponding time traces for the polarization decay
are obtained by Fourier transform as shown in Fig. 7.5c. The associated SPP lifetimes
can then either be directly deduced from the 1/e value of the maximum amplitude,
or obtained from the FWHM (Γ ) spectral line width by T 2 Γ = 2. Note that the
deviation from an ideal Lorentzian spectral or exponential time behavior in this model
calculation is due to the use of the experimentally measured dielectric function ε(ω)
as an input parameter and associated deviations from the ideal Drude behavior. The
resulting plasmon lifetimes are then shown in Fig. 7.5d as a function of resonant
energy (or aspect ratio). The dephasing times are found to be in the range of 18–
22 fs for energies between 1.0 and 1.7 eV, i.e., the free electron regime. The dramatic
decrease in lifetime at 2 eV is associated with the onset of the interband transition.
The interpretation of the dephasing time is complicated by the various possible
mechanisms contributing to the loss of phase coherence in the plasmon oscillation.
In general, the measured dephasing time T 2 is related to a population relaxation time
T 1 of participating quantum states, and pure dephasing T ≥
2 by 1/T 2 = 1/2T 1 +1/T ≥
2 .
The pure dephasing contribution T ≥
2 corresponds to elastic collisions of electrons,
which destroy only the phase coherence. Because of the high carrier density and
high electron scattering rate, this is expected to be negligible for SPPs. However,
this relation has limited applicability in this case. The SPP classical polarization
is described in terms of T 2 , i.e. the polarization decay through inelastic electron
scattering processes, but there is no associated population prior to the decay into
electron-hole pairs after decoherence. Instead, the underlying momentum scattering,
which gives rise to the loss in phase coherence, can be associated with the Drude
scattering τ D , with T 2 ∝ 2τ D .
Drude scattering leads to electron-hole excitation and corresponding absorption
effects, alternatively described via ohmic loss. It competes with radiative decay. The
larger effective oscillator size and polarization with increasing particle size leads to an
increase in the radiation damping contribution. For particles larger than ∝20 nm this
produces a monotonic trend of decreasing dephasing time with increasing particle
diameter.
The quasistatic ellipsoid model discussed above and shown in Fig. 7.5 neglects
radiative decay and thus provides only an upper limit for the dephasing time. In order
to account for radiation damping, we use the rigorous solution for the scattering of
a particle given by Mie theory [14, 11]. The scattering and extinction efficiencies
for the m-th multipole order is related to the scattering and extinction cross sections
σ sca,m and σ ext,m , and the geometrical cross section G = π R 2 , by
Q sca,m =
σ sca,m
G =
2
x 2 (2m + 1)(|a m |
2
+ |b m |
2
), and
Q ext,m =
σ ext,m
G =
2
x 2 (2m + 1)Re(a m + b m ),
(7.14)
with x = k R = ωn d (ω)R/c. The scattering coefficients a n and b n are given by
M. B. Raschke et al.
From the calculated spectra for a Au sphere or spheroids with different aspect
ratios as shown in Fig. 7.5b, the corresponding time traces for the polarization decay
are obtained by Fourier transform as shown in Fig. 7.5c. The associated SPP lifetimes
can then either be directly deduced from the 1/e value of the maximum amplitude,
or obtained from the FWHM (Γ ) spectral line width by T 2 Γ = 2. Note that the
deviation from an ideal Lorentzian spectral or exponential time behavior in this model
calculation is due to the use of the experimentally measured dielectric function ε(ω)
as an input parameter and associated deviations from the ideal Drude behavior. The
resulting plasmon lifetimes are then shown in Fig. 7.5d as a function of resonant
energy (or aspect ratio). The dephasing times are found to be in the range of 18–
22 fs for energies between 1.0 and 1.7 eV, i.e., the free electron regime. The dramatic
decrease in lifetime at 2 eV is associated with the onset of the interband transition.
The interpretation of the dephasing time is complicated by the various possible
mechanisms contributing to the loss of phase coherence in the plasmon oscillation.
In general, the measured dephasing time T 2 is related to a population relaxation time
T 1 of participating quantum states, and pure dephasing T ≥
2 by 1/T 2 = 1/2T 1 +1/T ≥
2 .
The pure dephasing contribution T ≥
2 corresponds to elastic collisions of electrons,
which destroy only the phase coherence. Because of the high carrier density and
high electron scattering rate, this is expected to be negligible for SPPs. However,
this relation has limited applicability in this case. The SPP classical polarization
is described in terms of T 2 , i.e. the polarization decay through inelastic electron
scattering processes, but there is no associated population prior to the decay into
electron-hole pairs after decoherence. Instead, the underlying momentum scattering,
which gives rise to the loss in phase coherence, can be associated with the Drude
scattering τ D , with T 2 ∝ 2τ D .
Drude scattering leads to electron-hole excitation and corresponding absorption
effects, alternatively described via ohmic loss. It competes with radiative decay. The
larger effective oscillator size and polarization with increasing particle size leads to an
increase in the radiation damping contribution. For particles larger than ∝20 nm this
produces a monotonic trend of decreasing dephasing time with increasing particle
diameter.
The quasistatic ellipsoid model discussed above and shown in Fig. 7.5 neglects
radiative decay and thus provides only an upper limit for the dephasing time. In order
to account for radiation damping, we use the rigorous solution for the scattering of
a particle given by Mie theory [14, 11]. The scattering and extinction efficiencies
for the m-th multipole order is related to the scattering and extinction cross sections
σ sca,m and σ ext,m , and the geometrical cross section G = π R 2 , by
Q sca,m =
σ sca,m
G =
2
x 2 (2m + 1)(|a m |
2
+ |b m |
2
), and
Q ext,m =
σ ext,m
G =
2
x 2 (2m + 1)Re(a m + b m ),
(7.14)
with x = k R = ωn d (ω)R/c. The scattering coefficients a n and b n are given by