7 Ultrafast and Nonlinear Plasmon Dynamics
241
band structure for Au ([Xe] 4 f 14 5d 10 6s) as an example, near the high-symmetry X
and Γ points. The dominant contributions to the interband d-sp transition are shown,
with onset at ∝1.9 eV and sharp rise at 2.4 eV.
The topology of the Fermi surface resembles the free electron sphere within
the first Brillouin zone, except for the 111 direction (L neck). Near the Fermi
level E F the optical absorption is weak due to the absence of direct transitions,
but is allowed for finite ω if translational symmetry is broken. This is the case
for electrons with momenta 2π/l, where l is the electron scattering length. With
increasing wavelength this gives rise to an increase in absorption. The effect of the
d band on the optical properties is discussed further in Sect. 7.1.6 after introducing
the free electron response.
7.1.5 Drude-Sommerfeld Model
Classically the motion of carriers in a metal can be described as ballistic under
the assumption of negligible Coulomb interaction. This is the so-called free electron
response of metals, described by the Drude-Sommerfeld model [7]. Damping, which
gives rise to ohmic resistance, can be introduced via the assumption of inelastic
and instantaneous collisions with unspecified scattering centers. This leaves as the
only key parameter the time τ D between collision events, defining a relaxation rate
Γ = 1/τ D . The equation of motion describing that relaxor behavior then corresponds
to that of a damped harmonic oscillator without a restoring force term, giving rise to
an apparent resonance at ω = 0 s −1 . This is the Drude peak, describing the increasing
absorption with decreasing frequency as mentioned above.
It is instructive to first consider the ideal Drude response without damping. From
the equation of motion of free carriers of density n subject only to a driving external
optical field E(t), the induced optical polarization is given by
P(t) = −nex(t) = −
ne 2
mω 2 E(t),
(7.6)
with x(t) the separation of electrons from the ions under the influence of the driving
field, and m and e the electron mass and charge, respectively. Based on that expression
the dielectric function of the charge plasma can be derived as
ε(ω) = 1 −
ω 2
p
ω 2 , with ω p =
ne 2
ε 0 m
(7.7)
termed the volume plasma frequency. This dielectric function is purely real, reflecting
the absence of an energy dissipating term. The conductivity σ (ω) is then purely
imaginary, indicating a 90 ◦ phase shift between an applied field and the induced
241
band structure for Au ([Xe] 4 f 14 5d 10 6s) as an example, near the high-symmetry X
and Γ points. The dominant contributions to the interband d-sp transition are shown,
with onset at ∝1.9 eV and sharp rise at 2.4 eV.
The topology of the Fermi surface resembles the free electron sphere within
the first Brillouin zone, except for the 111 direction (L neck). Near the Fermi
level E F the optical absorption is weak due to the absence of direct transitions,
but is allowed for finite ω if translational symmetry is broken. This is the case
for electrons with momenta 2π/l, where l is the electron scattering length. With
increasing wavelength this gives rise to an increase in absorption. The effect of the
d band on the optical properties is discussed further in Sect. 7.1.6 after introducing
the free electron response.
7.1.5 Drude-Sommerfeld Model
Classically the motion of carriers in a metal can be described as ballistic under
the assumption of negligible Coulomb interaction. This is the so-called free electron
response of metals, described by the Drude-Sommerfeld model [7]. Damping, which
gives rise to ohmic resistance, can be introduced via the assumption of inelastic
and instantaneous collisions with unspecified scattering centers. This leaves as the
only key parameter the time τ D between collision events, defining a relaxation rate
Γ = 1/τ D . The equation of motion describing that relaxor behavior then corresponds
to that of a damped harmonic oscillator without a restoring force term, giving rise to
an apparent resonance at ω = 0 s −1 . This is the Drude peak, describing the increasing
absorption with decreasing frequency as mentioned above.
It is instructive to first consider the ideal Drude response without damping. From
the equation of motion of free carriers of density n subject only to a driving external
optical field E(t), the induced optical polarization is given by
P(t) = −nex(t) = −
ne 2
mω 2 E(t),
(7.6)
with x(t) the separation of electrons from the ions under the influence of the driving
field, and m and e the electron mass and charge, respectively. Based on that expression
the dielectric function of the charge plasma can be derived as
ε(ω) = 1 −
ω 2
p
ω 2 , with ω p =
ne 2
ε 0 m
(7.7)
termed the volume plasma frequency. This dielectric function is purely real, reflecting
the absence of an energy dissipating term. The conductivity σ (ω) is then purely
imaginary, indicating a 90 ◦ phase shift between an applied field and the induced