5 Ultrafast Nonlinear Plasmonics
187
The change of the surface term ωλ S is due to the electron temperature dependence of g that can be estimated using Eq. 5.7 (Fig. 5.1). It yields a relatively small
contribution to ωε in
2 for not too small nanoparticles, as illustrated for 10 nm diameter gold nanospheres (inset of Fig. 5.5f, using Eqs. 5.6 and 5.31). As it increases
with the particle size (Eq. 5.6), it can play a significant role in small nanoparticles as
shown in the case of gold nanospheres [60]. It is the first manifestation of quantum
confinement in their nonlinear optical response.
5.4.2 Lattice Contribution
In the equilibrium regime (T e = T L = T eq ), the electronic mechanisms are almost
negligible, as ωT eq << ωT me
e
and most of the energy is in the lattice (Fig. 5.4).
ωε then reflects lattice temperature rise and the concomitant metal dilation. Note
that the latter effect, that impacts both the interband contribution, via electronic band
shifting (i.e., deformation potential coupling), and the intraband one, via change of
the electronic density, has permitted detection of coherent vibration of metal films
and nanoparticles following the concomitant modulation of their volume [33, 86,
88, 124]. All the induced changes are proportional to ωT L and thus rise with lattice
heating by the electrons (Fig. 5.4) and decay with metal cooling to its environment.
5.4.2.1 Interband Dielectric Function
Due to lattice anharmonicity, lattice heating leads to increase of the equilibrium
inter-atomic distance a L and to shift of the Fermi energy. This induces modification
of the electronic band structure, shifting the energy position of the electronic bands
and their filling, and thus of the metal interband absorption spectrum. ε ib is thus
modified, proportionally to ωT L . Assuming rigid band shift one gets:
ωε
ib
(ω) = −
γε ib
γ ω
ω
γ ω ib
γ T L
T 0
ωT L .
(5.33)
Displacement of the interband transition threshold ω ib with the T L can be estimated from calculation of the Fermi energy and the electronic band structure of
metal [80, 84]. For gold, one obtains around the L point of the Brillouin zone
(γ ω ib /γ T L ) T 0 ≈ −5 × 10 −5 eV/K comparable but smaller than the experimentally estimated value: (γ ω ib /γ T L ) T 0 ≈ −1.7 × 10 −4 eV/K [82]. A different shift
is estimated for the X point threshold, an effect that will not be included here as
contribution of the X point transitions to ε ib and ωε ib is smaller than for the L point.
The estimated changes of ε ib are shown in Fig. 5.5c,d using the above experimental
(γ ω ib /γ T L ) T 0 and the (γε/γ ω) values computed from the ε data of Johnson and
Christy. As expected, they show structures around the interband transition threshold
and rise with the lattice temperature.
187
The change of the surface term ωλ S is due to the electron temperature dependence of g that can be estimated using Eq. 5.7 (Fig. 5.1). It yields a relatively small
contribution to ωε in
2 for not too small nanoparticles, as illustrated for 10 nm diameter gold nanospheres (inset of Fig. 5.5f, using Eqs. 5.6 and 5.31). As it increases
with the particle size (Eq. 5.6), it can play a significant role in small nanoparticles as
shown in the case of gold nanospheres [60]. It is the first manifestation of quantum
confinement in their nonlinear optical response.
5.4.2 Lattice Contribution
In the equilibrium regime (T e = T L = T eq ), the electronic mechanisms are almost
negligible, as ωT eq << ωT me
e
and most of the energy is in the lattice (Fig. 5.4).
ωε then reflects lattice temperature rise and the concomitant metal dilation. Note
that the latter effect, that impacts both the interband contribution, via electronic band
shifting (i.e., deformation potential coupling), and the intraband one, via change of
the electronic density, has permitted detection of coherent vibration of metal films
and nanoparticles following the concomitant modulation of their volume [33, 86,
88, 124]. All the induced changes are proportional to ωT L and thus rise with lattice
heating by the electrons (Fig. 5.4) and decay with metal cooling to its environment.
5.4.2.1 Interband Dielectric Function
Due to lattice anharmonicity, lattice heating leads to increase of the equilibrium
inter-atomic distance a L and to shift of the Fermi energy. This induces modification
of the electronic band structure, shifting the energy position of the electronic bands
and their filling, and thus of the metal interband absorption spectrum. ε ib is thus
modified, proportionally to ωT L . Assuming rigid band shift one gets:
ωε
ib
(ω) = −
γε ib
γ ω
ω
γ ω ib
γ T L
T 0
ωT L .
(5.33)
Displacement of the interband transition threshold ω ib with the T L can be estimated from calculation of the Fermi energy and the electronic band structure of
metal [80, 84]. For gold, one obtains around the L point of the Brillouin zone
(γ ω ib /γ T L ) T 0 ≈ −5 × 10 −5 eV/K comparable but smaller than the experimentally estimated value: (γ ω ib /γ T L ) T 0 ≈ −1.7 × 10 −4 eV/K [82]. A different shift
is estimated for the X point threshold, an effect that will not be included here as
contribution of the X point transitions to ε ib and ωε ib is smaller than for the L point.
The estimated changes of ε ib are shown in Fig. 5.5c,d using the above experimental
(γ ω ib /γ T L ) T 0 and the (γε/γ ω) values computed from the ε data of Johnson and
Christy. As expected, they show structures around the interband transition threshold
and rise with the lattice temperature.