5 Ultrafast Nonlinear Plasmonics
171
not too small particle, a continuum of conduction band states is recovered and one
obtains [26, 76]:
g(ω, T e ) =
1
ω E 2
F
∞
0
E
3/2
√
E + ω f (E)[1 − f (E + ω)]d E.
(5.7)
For noble metals, g ≈ 0.7 at room temperature in the optical domain (g varies
weakly with frequency), consistent with experimental results obtained in single
nanoparticles [73]. Note that, for not too small particles, λ nano shows a 1/D correction as compared to λ bulk [73]. This is the same size dependence as the surface
term (Eq. 5.6) and the two effects cannot be distinguished so that λ n can be rewritten:
λ n (ω, T L , T e ) = λ bulk (ω, T L , T e ) + 2g e f f (ω, T e )v F /D,
(5.8)
where g e f f is a modified g factor lumping all confinement effects.
In noble metal, in the visible range, ε ib is associated to interband transitions from
the full d-bands to empty states, above the Fermi energy, E F , in the conduction
band, and to absorption from latter to empty band states of higher energy. The formers dominate with a threshold at ω ib ≈ 4.1eV and 1.9 eV in silver and gold,
respectively, interband absorption increasing more importantly above about 2.3 eV
in gold (Fig. 5.1). The absorption rise in the vicinity of ω ib can be well reproduced in
gold and silver using the models developed for thermomodulation studies [77–79].
In gold, only d-band to conduction band (d → c) transitions are taking place close
to ω ib , with a dominant contribution around the L point of the Brillouin zone and a
weaker one at lower energy around the X point. In the constant transition matrix element approximation and neglecting the width of the electronic states, the interband
contribution to the imaginary part of the gold dielectric function can then be written:
ε
ib
2 (ω) =
A
ω 2
E M
dc,L
E m
dc,L
D
L
d→c (E, ω) f (E)d E − K
2
X L
E M
dc,X
E m
dc,X
D
X
d→c (E, ω) f (E)d E
.
(5.9)
D
L ,X
d→c (E, ω) is the energy dependent joint density of state for the d → c transition
around the L and X points as given in [78] using band structure calculations [80].
K X L is their relative amplitude. The two parameters, K X L and the amplitude A, are
obtained by fitting the experimental ε ib
2 values deduced by subtracting the Drude
contribution to the measured ε bulk
2
, Eq. 5.1 [38, 65, 81, 82]. Different value of K X L
are obtained using different set of measured ε bulk
2
. For the data of [81] one obtains
K X L ≈ 0.37 (Fig. 5.1), while for the data of [83] K X L ≈ 0.84 (note that these
different values yield similar nonlinear spectra, the X point influence on ωε ib being
weak). A similar approach can be used for silver, absorption mostly taking place
around the L point of the Brillouin zone, with a main contribution due to transitions
from the d-bands to the conduction band and, a weaker one, due to transition from
the conduction band to a higher energy empty s-band [77, 79] as shown by band
structure modeling [84].
171
not too small particle, a continuum of conduction band states is recovered and one
obtains [26, 76]:
g(ω, T e ) =
1
ω E 2
F
∞
0
E
3/2
√
E + ω f (E)[1 − f (E + ω)]d E.
(5.7)
For noble metals, g ≈ 0.7 at room temperature in the optical domain (g varies
weakly with frequency), consistent with experimental results obtained in single
nanoparticles [73]. Note that, for not too small particles, λ nano shows a 1/D correction as compared to λ bulk [73]. This is the same size dependence as the surface
term (Eq. 5.6) and the two effects cannot be distinguished so that λ n can be rewritten:
λ n (ω, T L , T e ) = λ bulk (ω, T L , T e ) + 2g e f f (ω, T e )v F /D,
(5.8)
where g e f f is a modified g factor lumping all confinement effects.
In noble metal, in the visible range, ε ib is associated to interband transitions from
the full d-bands to empty states, above the Fermi energy, E F , in the conduction
band, and to absorption from latter to empty band states of higher energy. The formers dominate with a threshold at ω ib ≈ 4.1eV and 1.9 eV in silver and gold,
respectively, interband absorption increasing more importantly above about 2.3 eV
in gold (Fig. 5.1). The absorption rise in the vicinity of ω ib can be well reproduced in
gold and silver using the models developed for thermomodulation studies [77–79].
In gold, only d-band to conduction band (d → c) transitions are taking place close
to ω ib , with a dominant contribution around the L point of the Brillouin zone and a
weaker one at lower energy around the X point. In the constant transition matrix element approximation and neglecting the width of the electronic states, the interband
contribution to the imaginary part of the gold dielectric function can then be written:
ε
ib
2 (ω) =
A
ω 2
E M
dc,L
E m
dc,L
D
L
d→c (E, ω) f (E)d E − K
2
X L
E M
dc,X
E m
dc,X
D
X
d→c (E, ω) f (E)d E
.
(5.9)
D
L ,X
d→c (E, ω) is the energy dependent joint density of state for the d → c transition
around the L and X points as given in [78] using band structure calculations [80].
K X L is their relative amplitude. The two parameters, K X L and the amplitude A, are
obtained by fitting the experimental ε ib
2 values deduced by subtracting the Drude
contribution to the measured ε bulk
2
, Eq. 5.1 [38, 65, 81, 82]. Different value of K X L
are obtained using different set of measured ε bulk
2
. For the data of [81] one obtains
K X L ≈ 0.37 (Fig. 5.1), while for the data of [83] K X L ≈ 0.84 (note that these
different values yield similar nonlinear spectra, the X point influence on ωε ib being
weak). A similar approach can be used for silver, absorption mostly taking place
around the L point of the Brillouin zone, with a main contribution due to transitions
from the d-bands to the conduction band and, a weaker one, due to transition from
the conduction band to a higher energy empty s-band [77, 79] as shown by band
structure modeling [84].